Equivalent Cauchy sequences

Analysis I, Section 5.2: Equivalent Cauchy sequences

I have attempted to make the translation as faithful a paraphrasing as possible of the original text. When there is a choice between a more idiomatic Lean solution and a more faithful translation, I have generally chosen the latter. In particular, there will be places where the Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided doing so.

Main constructions and results of this section:

  • Notion of an ε-close and eventually ε-close sequences of rationals.

  • Notion of an equivalent Cauchy sequence of rationals.

Tips from past users

Users of the companion who have completed the exercises in this section are welcome to send their tips for future users in this section as PRs.

  • (Add tip here)

abbrev Rat.CloseSeq (ε: ℚ) (a b: Chapter5.Sequence) : Prop := ∀ n, n ≥ a.n₀ → n ≥ b.n₀ → ε.Close (a n) (b n)abbrev Rat.EventuallyClose (ε: ℚ) (a b: Chapter5.Sequence) : Prop := ∃ N, ε.CloseSeq (a.from N) (b.from N)namespace Chapter5

Definition 5.2.1 ($ε$-close sequences)

lemma Rat.closeSeq_def (ε: ℚ) (a b: Sequence) : ε.CloseSeq a b ↔ ∀ n, n ≥ a.n₀ → n ≥ b.n₀ → ε.Close (a n) (b n) := ε:ℚa:Sequenceb:Sequence⊢ ε.CloseSeq a b ↔ ∀ n ≥ a.n₀, n ≥ b.n₀ → ε.Close (a.seq n) (b.seq n) All goals completed! 🐙

Example 5.2.2

declaration uses `sorry`example : (0.1:ℚ).CloseSeq ((fun n:ℕ ↦ ((-1)^n:ℚ)):Sequence) ((fun n:ℕ ↦ ((1.1:ℚ) * (-1)^n)):Sequence) := ⊢ Rat.CloseSeq 0.1 (↑fun n => (-1) ^ n) ↑fun n => 1.1 * (-1) ^ n All goals completed! 🐙

Example 5.2.2

declaration uses `sorry`example : ¬ (0.1:ℚ).Steady ((fun n:ℕ ↦ ((-1)^n:ℚ)):Sequence) := ⊢ ¬Rat.Steady 0.1 ↑fun n => (-1) ^ n All goals completed! 🐙

Example 5.2.2

declaration uses `sorry`example : ¬ (0.1:ℚ).Steady ((fun n:ℕ ↦ ((1.1:ℚ) * (-1)^n)):Sequence) := ⊢ ¬Rat.Steady 0.1 ↑fun n => 1.1 * (-1) ^ n All goals completed! 🐙

Definition 5.2.3 (Eventually ε-close sequences)

lemma Rat.eventuallyClose_def (ε: ℚ) (a b: Sequence) : ε.EventuallyClose a b ↔ ∃ N, ε.CloseSeq (a.from N) (b.from N) := ε:ℚa:Sequenceb:Sequence⊢ ε.EventuallyClose a b ↔ ∃ N, ε.CloseSeq (a.from N) (b.from N) All goals completed! 🐙

Definition 5.2.3 (Eventually ε-close sequences)

lemma declaration uses `sorry`Rat.eventuallyClose_iff (ε: ℚ) (a b: ℕ → ℚ) : ε.EventuallyClose (a:Sequence) (b:Sequence) ↔ ∃ N, ∀ n ≥ N, |a n - b n| ≤ ε := ε:ℚa:ℕ → ℚb:ℕ → ℚ⊢ ε.EventuallyClose ↑a ↑b ↔ ∃ N, ∀ n ≥ N, |a n - b n| ≤ ε All goals completed! 🐙

Example 5.2.5

declaration uses `sorry`example : ¬ (0.1:ℚ).CloseSeq ((fun n:ℕ ↦ (1:ℚ)+10^(-(n:ℤ)-1)):Sequence) ((fun n:ℕ ↦ (1:ℚ)-10^(-(n:ℤ)-1)):Sequence) := ⊢ ¬Rat.CloseSeq 0.1 (↑fun n => 1 + 10 ^ (-↑n - 1)) ↑fun n => 1 - 10 ^ (-↑n - 1) All goals completed! 🐙
declaration uses `sorry`example : (0.1:ℚ).EventuallyClose ((fun n:ℕ ↦ (1:ℚ)+10^(-(n:ℤ)-1)):Sequence) ((fun n:ℕ ↦ (1:ℚ)-10^(-(n:ℤ)-1)):Sequence) := ⊢ Rat.EventuallyClose 0.1 (↑fun n => 1 + 10 ^ (-↑n - 1)) ↑fun n => 1 - 10 ^ (-↑n - 1) All goals completed! 🐙declaration uses `sorry`example : (0.01:ℚ).EventuallyClose ((fun n:ℕ ↦ (1:ℚ)+10^(-(n:ℤ)-1)):Sequence) ((fun n:ℕ ↦ (1:ℚ)-10^(-(n:ℤ)-1)):Sequence) := ⊢ Rat.EventuallyClose 1e-2 (↑fun n => 1 + 10 ^ (-↑n - 1)) ↑fun n => 1 - 10 ^ (-↑n - 1) All goals completed! 🐙

Definition 5.2.6 (Equivalent sequences)

abbrev Sequence.Equiv (a b: ℕ → ℚ) : Prop := ∀ ε > (0:ℚ), ε.EventuallyClose (a:Sequence) (b:Sequence)

Definition 5.2.6 (Equivalent sequences)

lemma Sequence.equiv_def (a b: ℕ → ℚ) : Equiv a b ↔ ∀ (ε:ℚ), ε > 0 → ε.EventuallyClose (a:Sequence) (b:Sequence) := a:ℕ → ℚb:ℕ → ℚ⊢ Equiv a b ↔ ∀ ε > 0, ε.EventuallyClose ↑a ↑b All goals completed! 🐙

Definition 5.2.6 (Equivalent sequences)

lemma declaration uses `sorry`Sequence.equiv_iff (a b: ℕ → ℚ) : Equiv a b ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, |a n - b n| ≤ ε := a:ℕ → ℚb:ℕ → ℚ⊢ Equiv a b ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, |a n - b n| ≤ ε All goals completed! 🐙

Proposition 5.2.8

lemma Sequence.equiv_example : -- This proof is perhaps more complicated than it needs to be; a shorter version may be -- possible that is still faithful to the original text. Equiv (fun n:ℕ ↦ (1:ℚ)+10^(-(n:ℤ)-1)) (fun n:ℕ ↦ (1:ℚ)-10^(-(n:ℤ)-1)) := ⊢ Equiv (fun n => 1 + 10 ^ (-↑n - 1)) fun n => 1 - 10 ^ (-↑n - 1) a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)⊢ Equiv a fun n => 1 - 10 ^ (-↑n - 1) a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)⊢ Equiv a b a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, |a n - b n| ≤ ε intro ε a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0⊢ ∃ N, ∀ n ≥ N, |a n - b n| ≤ ε have hab (n:ℕ) : |a n - b n| = 2 * 10 ^ (-(n:ℤ)-1) := calc _ = |((1:ℚ) + (10:ℚ)^(-(n:ℤ)-1)) - ((1:ℚ) - (10:ℚ)^(-(n:ℤ)-1))| := rfl _ = |2 * (10:ℚ)^(-(n:ℤ)-1)| := a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0n:ℕ⊢ |1 + 10 ^ (-↑n - 1) - (1 - 10 ^ (-↑n - 1))| = |2 * 10 ^ (-↑n - 1)| All goals completed! 🐙 _ = _ := abs_of_nonneg (a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0n:ℕ⊢ 0 ≤ 2 * 10 ^ (-↑n - 1) All goals completed! 🐙) have hab' (N:ℕ) : ∀ n ≥ N, |a n - b n| ≤ 2 * 10 ^(-(N:ℤ)-1) := ⊢ Equiv (fun n => 1 + 10 ^ (-↑n - 1)) fun n => 1 - 10 ^ (-↑n - 1) intro n a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))N:ℕn:ℕhn:n ≥ N⊢ |a n - b n| ≤ 2 * 10 ^ (-↑N - 1); a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))N:ℕn:ℕhn:n ≥ N⊢ 2 * 10 ^ (-↑n - 1) ≤ 2 * 10 ^ (-↑N - 1); a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))N:ℕn:ℕhn:n ≥ N⊢ 1 ≤ 10; All goals completed! 🐙 have hN : ∃ N:ℕ, 2 * (10:ℚ) ^(-(N:ℤ)-1) ≤ ε := ⊢ Equiv (fun n => 1 + 10 ^ (-↑n - 1)) fun n => 1 - 10 ^ (-↑n - 1) have hN' (N:ℕ) : 2 * (10:ℚ)^(-(N:ℤ)-1) ≤ 2/(N+1) := calc _ = 2 / (10:ℚ)^(N+1) := a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ 2 * 10 ^ (-↑N - 1) = 2 / 10 ^ (N + 1) a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ 10 ^ (-↑N - 1) * 10 ^ (N + 1) = 1 All goals completed! 🐙 _ ≤ _ := a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ 2 / 10 ^ (N + 1) ≤ 2 / (↑N + 1) a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ ↑N + 1 ≤ 10 ^ (N + 1) apply le_trans _ (pow_le_pow_left₀ (show 0 ≤ (2:ℚ) a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ 2 / 10 ^ (N + 1) ≤ 2 / (↑N + 1) All goals completed! 🐙) (show (2:ℚ) ≤ 10 a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ 2 / 10 ^ (N + 1) ≤ 2 / (↑N + 1) All goals completed! 🐙) _) a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ ↑N + 1 = ↑(N + 1)a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ 2 ^ (N + 1) = ↑(2 ^ (N + 1))a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ AddMonoidWithOne ℚa:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ AddLeftMono ℚa:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ ZeroLEOneClass ℚa:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ CharZero ℚ a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ ↑N + 1 = ↑(N + 1)a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ 2 ^ (N + 1) = ↑(2 ^ (N + 1))a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ AddMonoidWithOne ℚa:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ AddLeftMono ℚa:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ ZeroLEOneClass ℚa:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕ⊢ CharZero ℚ try All goals completed! 🐙 all_goals All goals completed! 🐙 a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))hN':∀ (N : ℕ), 2 * 10 ^ (-↑N - 1) ≤ 2 / (↑N + 1) := fun N => Trans.trans (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.trans (Eq.refl (10 ^ (-↑N - 1))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (10 ^ (-↑N - 1)))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, 10 ^ (-↑N - 1)) (mul_one ((1, 2) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (N + 1))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (10 ^ (-↑N - 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (10 ^ (N + 1)) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (10 ^ (-↑N - 1)) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 10 ^ (N + 1)) ((1, 10 ^ (-↑N - 1)) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = 10 ^ (-↑N - 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 10 ^ (-↑N - 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 10 ^ (-↑N - 1)).2 (1, 10 ^ (-↑N - 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (10 ^ (-↑N - 1))))) (Mathlib.Tactic.FieldSimp.zpow'_one (10 ^ (N + 1))))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.trans (Eq.refl (10 ^ (N + 1))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (10 ^ (N + 1)))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃ (1, 10 ^ (N + 1)) (div_one ((1, 2) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (10 ^ (N + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (N + 1))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero 1 (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Eq.refl false)) one_ne_zero)))) (of_eq_true (Eq.trans (congrFun' (congrArg Eq (Eq.trans (Eq.trans (congrArg (HMul.hMul (10 ^ (-↑N - 1))) (Eq.trans (equiv_example._simp_1 10 (N + 1)) (congrArg (HPow.hPow 10) (Eq.trans (Nat.cast_add N 1) (congrArg (HAdd.hAdd ↑N) Nat.cast_one))))) ((fun m n => Eq.symm (zpow_add₀ (have this := Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Eq.refl false); this) m n)) (-↑N - 1) (↑N + 1))) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (sub_add_add_cancel (-↑N) (↑N) 1) (neg_add_cancel ↑N))) (zpow_zero 10)))) 1) (eq_self 1)))) (div_le_div₀ (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))) (le_refl 2) (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' N) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1)))) (le_trans (Eq.mpr (eq_of_heq ((fun α self self' e'_2 a a' e'_3 a_1 a'_1 e'_4 => Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a'_1)) e'_2 (fun h => Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a'_1)) (fun e_2 h => Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a'_1)) e'_3 (fun h => Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a'_1)) (fun e_3 h => Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_4 ≍ x → (a ≤ a_1) ≍ (a ≤ a'_1)) e'_4 (fun h => Eq.ndrec (motive := fun a' => ∀ (e_4 : a_1 = a'), e_4 ≍ Eq.refl a_1 → (a ≤ a_1) ≍ (a ≤ a')) (fun e_4 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_4) (Eq.refl a'_1) (HEq.refl e'_4)) (Eq.symm h) e'_3) (Eq.refl a') (HEq.refl e'_3)) (Eq.symm h) e'_2) (Eq.refl self') (HEq.refl e'_2)) ℚ Rat.instPreorder.toLE Rat.instPartialOrder.toLE (Eq.refl Rat.instPreorder.toLE) (↑N + 1) (↑(N + 1)) (of_eq_true (Eq.trans (congrArg (Eq (↑N + 1)) (Eq.trans (Nat.cast_add N 1) (congrArg (HAdd.hAdd ↑N) Nat.cast_one))) (eq_self (↑N + 1)))) (2 ^ (N + 1)) (↑(2 ^ (N + 1))) (of_eq_true (Eq.trans (congrArg (Eq (2 ^ (N + 1))) (Nat.cast_pow 2 (N + 1))) (eq_self (2 ^ (N + 1))))))) (Nat.cast_le.mpr (Section_4_3.two_pow_geq (N + 1)))) (pow_le_pow_left₀ (have this := Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl true); this) (have this := Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true); this) (N + 1))))N:ℕhN:2 / ε < ↑N⊢ ∃ N, 2 * 10 ^ (-↑N - 1) ≤ ε a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))hN':∀ (N : ℕ), 2 * 10 ^ (-↑N - 1) ≤ 2 / (↑N + 1) := fun N => Trans.trans (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.trans (Eq.refl (10 ^ (-↑N - 1))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (10 ^ (-↑N - 1)))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, 10 ^ (-↑N - 1)) (mul_one ((1, 2) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (N + 1))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (10 ^ (-↑N - 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (10 ^ (N + 1)) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (10 ^ (-↑N - 1)) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 10 ^ (N + 1)) ((1, 10 ^ (-↑N - 1)) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = 10 ^ (-↑N - 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 10 ^ (-↑N - 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 10 ^ (-↑N - 1)).2 (1, 10 ^ (-↑N - 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (10 ^ (-↑N - 1))))) (Mathlib.Tactic.FieldSimp.zpow'_one (10 ^ (N + 1))))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.trans (Eq.refl (10 ^ (N + 1))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (10 ^ (N + 1)))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃ (1, 10 ^ (N + 1)) (div_one ((1, 2) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (10 ^ (N + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (N + 1))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero 1 (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Eq.refl false)) one_ne_zero)))) (of_eq_true (Eq.trans (congrFun' (congrArg Eq (Eq.trans (Eq.trans (congrArg (HMul.hMul (10 ^ (-↑N - 1))) (Eq.trans (equiv_example._simp_1 10 (N + 1)) (congrArg (HPow.hPow 10) (Eq.trans (Nat.cast_add N 1) (congrArg (HAdd.hAdd ↑N) Nat.cast_one))))) ((fun m n => Eq.symm (zpow_add₀ (have this := Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Eq.refl false); this) m n)) (-↑N - 1) (↑N + 1))) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (sub_add_add_cancel (-↑N) (↑N) 1) (neg_add_cancel ↑N))) (zpow_zero 10)))) 1) (eq_self 1)))) (div_le_div₀ (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))) (le_refl 2) (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' N) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1)))) (le_trans (Eq.mpr (eq_of_heq ((fun α self self' e'_2 a a' e'_3 a_1 a'_1 e'_4 => Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a'_1)) e'_2 (fun h => Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a'_1)) (fun e_2 h => Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a'_1)) e'_3 (fun h => Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a'_1)) (fun e_3 h => Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_4 ≍ x → (a ≤ a_1) ≍ (a ≤ a'_1)) e'_4 (fun h => Eq.ndrec (motive := fun a' => ∀ (e_4 : a_1 = a'), e_4 ≍ Eq.refl a_1 → (a ≤ a_1) ≍ (a ≤ a')) (fun e_4 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_4) (Eq.refl a'_1) (HEq.refl e'_4)) (Eq.symm h) e'_3) (Eq.refl a') (HEq.refl e'_3)) (Eq.symm h) e'_2) (Eq.refl self') (HEq.refl e'_2)) ℚ Rat.instPreorder.toLE Rat.instPartialOrder.toLE (Eq.refl Rat.instPreorder.toLE) (↑N + 1) (↑(N + 1)) (of_eq_true (Eq.trans (congrArg (Eq (↑N + 1)) (Eq.trans (Nat.cast_add N 1) (congrArg (HAdd.hAdd ↑N) Nat.cast_one))) (eq_self (↑N + 1)))) (2 ^ (N + 1)) (↑(2 ^ (N + 1))) (of_eq_true (Eq.trans (congrArg (Eq (2 ^ (N + 1))) (Nat.cast_pow 2 (N + 1))) (eq_self (2 ^ (N + 1))))))) (Nat.cast_le.mpr (Section_4_3.two_pow_geq (N + 1)))) (pow_le_pow_left₀ (have this := Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl true); this) (have this := Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true); this) (N + 1))))N:ℕhN:2 / ε < ↑N⊢ 2 / (↑N + 1) ≤ ε a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))hN':∀ (N : ℕ), 2 * 10 ^ (-↑N - 1) ≤ 2 / (↑N + 1) := fun N => Trans.trans (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.trans (Eq.refl (10 ^ (-↑N - 1))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (10 ^ (-↑N - 1)))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, 10 ^ (-↑N - 1)) (mul_one ((1, 2) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (N + 1))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (10 ^ (-↑N - 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (10 ^ (N + 1)) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (10 ^ (-↑N - 1)) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 10 ^ (N + 1)) ((1, 10 ^ (-↑N - 1)) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = 10 ^ (-↑N - 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 10 ^ (-↑N - 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 10 ^ (-↑N - 1)).2 (1, 10 ^ (-↑N - 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (10 ^ (-↑N - 1))))) (Mathlib.Tactic.FieldSimp.zpow'_one (10 ^ (N + 1))))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.trans (Eq.refl (10 ^ (N + 1))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (10 ^ (N + 1)))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃ (1, 10 ^ (N + 1)) (div_one ((1, 2) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (10 ^ (N + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (N + 1))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero 1 (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Eq.refl false)) one_ne_zero)))) (of_eq_true (Eq.trans (congrFun' (congrArg Eq (Eq.trans (Eq.trans (congrArg (HMul.hMul (10 ^ (-↑N - 1))) (Eq.trans (equiv_example._simp_1 10 (N + 1)) (congrArg (HPow.hPow 10) (Eq.trans (Nat.cast_add N 1) (congrArg (HAdd.hAdd ↑N) Nat.cast_one))))) ((fun m n => Eq.symm (zpow_add₀ (have this := Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Eq.refl false); this) m n)) (-↑N - 1) (↑N + 1))) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (sub_add_add_cancel (-↑N) (↑N) 1) (neg_add_cancel ↑N))) (zpow_zero 10)))) 1) (eq_self 1)))) (div_le_div₀ (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))) (le_refl 2) (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' N) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1)))) (le_trans (Eq.mpr (eq_of_heq ((fun α self self' e'_2 a a' e'_3 a_1 a'_1 e'_4 => Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a'_1)) e'_2 (fun h => Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a'_1)) (fun e_2 h => Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a'_1)) e'_3 (fun h => Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a'_1)) (fun e_3 h => Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_4 ≍ x → (a ≤ a_1) ≍ (a ≤ a'_1)) e'_4 (fun h => Eq.ndrec (motive := fun a' => ∀ (e_4 : a_1 = a'), e_4 ≍ Eq.refl a_1 → (a ≤ a_1) ≍ (a ≤ a')) (fun e_4 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_4) (Eq.refl a'_1) (HEq.refl e'_4)) (Eq.symm h) e'_3) (Eq.refl a') (HEq.refl e'_3)) (Eq.symm h) e'_2) (Eq.refl self') (HEq.refl e'_2)) ℚ Rat.instPreorder.toLE Rat.instPartialOrder.toLE (Eq.refl Rat.instPreorder.toLE) (↑N + 1) (↑(N + 1)) (of_eq_true (Eq.trans (congrArg (Eq (↑N + 1)) (Eq.trans (Nat.cast_add N 1) (congrArg (HAdd.hAdd ↑N) Nat.cast_one))) (eq_self (↑N + 1)))) (2 ^ (N + 1)) (↑(2 ^ (N + 1))) (of_eq_true (Eq.trans (congrArg (Eq (2 ^ (N + 1))) (Nat.cast_pow 2 (N + 1))) (eq_self (2 ^ (N + 1))))))) (Nat.cast_le.mpr (Section_4_3.two_pow_geq (N + 1)))) (pow_le_pow_left₀ (have this := Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl true); this) (have this := Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true); this) (N + 1))))N:ℕhN:2 / ε < ↑N⊢ 2 ≤ ε * (↑N + 1) a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))hN':∀ (N : ℕ), 2 * 10 ^ (-↑N - 1) ≤ 2 / (↑N + 1) := fun N => Trans.trans (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.trans (Eq.refl (10 ^ (-↑N - 1))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (10 ^ (-↑N - 1)))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, 10 ^ (-↑N - 1)) (mul_one ((1, 2) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (N + 1))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (10 ^ (-↑N - 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (10 ^ (N + 1)) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (10 ^ (-↑N - 1)) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 10 ^ (N + 1)) ((1, 10 ^ (-↑N - 1)) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = 10 ^ (-↑N - 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 10 ^ (-↑N - 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 10 ^ (-↑N - 1)).2 (1, 10 ^ (-↑N - 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (10 ^ (-↑N - 1))))) (Mathlib.Tactic.FieldSimp.zpow'_one (10 ^ (N + 1))))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.trans (Eq.refl (10 ^ (N + 1))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (10 ^ (N + 1)))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval₃ (1, 10 ^ (N + 1)) (div_one ((1, 2) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (10 ^ (N + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (N + 1))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero 1 (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Eq.refl false)) one_ne_zero)))) (of_eq_true (Eq.trans (congrFun' (congrArg Eq (Eq.trans (Eq.trans (congrArg (HMul.hMul (10 ^ (-↑N - 1))) (Eq.trans (equiv_example._simp_1 10 (N + 1)) (congrArg (HPow.hPow 10) (Eq.trans (Nat.cast_add N 1) (congrArg (HAdd.hAdd ↑N) Nat.cast_one))))) ((fun m n => Eq.symm (zpow_add₀ (have this := Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Eq.refl false); this) m n)) (-↑N - 1) (↑N + 1))) (Eq.trans (congrArg (HPow.hPow 10) (Eq.trans (sub_add_add_cancel (-↑N) (↑N) 1) (neg_add_cancel ↑N))) (zpow_zero 10)))) 1) (eq_self 1)))) (div_le_div₀ (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))) (le_refl 2) (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' N) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1)))) (le_trans (Eq.mpr (eq_of_heq ((fun α self self' e'_2 a a' e'_3 a_1 a'_1 e'_4 => Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a'_1)) e'_2 (fun h => Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a'_1)) (fun e_2 h => Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a'_1)) e'_3 (fun h => Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a'_1)) (fun e_3 h => Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_4 ≍ x → (a ≤ a_1) ≍ (a ≤ a'_1)) e'_4 (fun h => Eq.ndrec (motive := fun a' => ∀ (e_4 : a_1 = a'), e_4 ≍ Eq.refl a_1 → (a ≤ a_1) ≍ (a ≤ a')) (fun e_4 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_4) (Eq.refl a'_1) (HEq.refl e'_4)) (Eq.symm h) e'_3) (Eq.refl a') (HEq.refl e'_3)) (Eq.symm h) e'_2) (Eq.refl self') (HEq.refl e'_2)) ℚ Rat.instPreorder.toLE Rat.instPartialOrder.toLE (Eq.refl Rat.instPreorder.toLE) (↑N + 1) (↑(N + 1)) (of_eq_true (Eq.trans (congrArg (Eq (↑N + 1)) (Eq.trans (Nat.cast_add N 1) (congrArg (HAdd.hAdd ↑N) Nat.cast_one))) (eq_self (↑N + 1)))) (2 ^ (N + 1)) (↑(2 ^ (N + 1))) (of_eq_true (Eq.trans (congrArg (Eq (2 ^ (N + 1))) (Nat.cast_pow 2 (N + 1))) (eq_self (2 ^ (N + 1))))))) (Nat.cast_le.mpr (Section_4_3.two_pow_geq (N + 1)))) (pow_le_pow_left₀ (have this := Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl true); this) (have this := Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true); this) (N + 1))))N:ℕhN:2 < ↑N * ε⊢ 2 ≤ ε * (↑N + 1) All goals completed! 🐙 a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕhN:2 * 10 ^ (-↑N - 1) ≤ ε⊢ ∃ N, ∀ n ≥ N, |a n - b n| ≤ ε; a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕhN:2 * 10 ^ (-↑N - 1) ≤ ε⊢ ∀ n ≥ N, |a n - b n| ≤ ε; intro n a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |a n - b n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg Eq (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (congrArg abs (Eq.trans (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf' (congrArg (HPow.hPow 10) (Eq.trans (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (Eq.trans (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow ↑n) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one ↑n))) (Eq.trans Mathlib.Tactic.RingNF.int_rawCast_neg (congrArg Neg.neg Mathlib.Tactic.RingNF.nat_rawCast_1))) (Mathlib.Tactic.RingNF.mul_neg (↑n) 1)) (congrArg Neg.neg (mul_one ↑n)))) 0) (add_zero (-↑n)))) (Mathlib.Tactic.RingNF.add_neg (-1) ↑n))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (10 ^ (-1 - ↑n)) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((10 ^ (-1 - ↑n)) ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Eq.trans (congrFun' (congrArg HAdd.hAdd (congrFun' (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) 2)) 0) (add_zero (10 ^ (-1 - ↑n) * 2)))))) (eq_self |10 ^ (-1 - ↑n) * 2|)))) (abs_of_nonneg (le_of_lt (mul_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))) (zpow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) (-↑n - 1)))))hab':∀ (N n : ℕ), n ≥ N → |a n - b n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (hab n))) (mul_le_mul_of_nonneg_left (zpow_le_zpow_right₀ (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 10)) (Eq.refl true)) (sub_le_sub_right (neg_le_neg (Nat.mono_cast hn)) 1)) (le_of_lt (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2)))))N:ℕhN:2 * 10 ^ (-↑N - 1) ≤ εn:ℕhn:n ≥ N⊢ |a n - b n| ≤ ε All goals completed! 🐙

Exercise 5.2.1

theorem declaration uses `sorry`Sequence.isCauchy_of_equiv {a b: ℕ → ℚ} (hab: Equiv a b) : (a:Sequence).IsCauchy ↔ (b:Sequence).IsCauchy := a:ℕ → ℚb:ℕ → ℚhab:Equiv a b⊢ (↑a).IsCauchy ↔ (↑b).IsCauchy All goals completed! 🐙

Exercise 5.2.2

theorem declaration uses `sorry`Sequence.isBounded_of_eventuallyClose {ε:ℚ} {a b: ℕ → ℚ} (hab: ε.EventuallyClose a b) : (a:Sequence).IsBounded ↔ (b:Sequence).IsBounded := ε:ℚa:ℕ → ℚb:ℕ → ℚhab:ε.EventuallyClose ↑a ↑b⊢ (↑a).IsBounded ↔ (↑b).IsBounded All goals completed! 🐙
end Chapter5