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.

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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 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) a:ℕ → ℚ := fun n => 1 + 10 ^ (-↑n - 1)b:ℕ → ℚ := fun n => 1 - 10 ^ (-↑n - 1)ε:ℚhε:ε > 0hab:∀ (n : ℕ), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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) * 10 ^ (N + 1) = 2 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (of_eq_true (Eq.trans (congr (congrArg (fun x => Eq |x|) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 (congrArg (fun x => x + 0) (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)))) (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 (congrArg (fun x => x * 2 + 0) (Eq.trans (congrArg (HPow.hPow (10 ^ (-1 - ↑n))) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one (10 ^ (-1 - ↑n))))) (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 → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N n hn => Eq.mpr (id (congrArg (fun _a => _a ≤ 2 * 10 ^ (-↑N - 1)) (@_fvar.6550 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 (GCongr.natCast_le_natCast 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))hab':∀ (N n : ℕ), n ≥ N → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N => @?_mvar.12521 NhN':∀ (N : ℕ), 2 * 10 ^ (-↑N - 1) ≤ 2 / (↑N + 1) := fun N => Trans.trans (@?_mvar.20396 N) (@?_mvar.20397 N)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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))hab':∀ (N n : ℕ), n ≥ N → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N => @?_mvar.12521 NhN':∀ (N : ℕ), 2 * 10 ^ (-↑N - 1) ≤ 2 / (↑N + 1) := fun N => Trans.trans (@?_mvar.20396 N) (@?_mvar.20397 N)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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))hab':∀ (N n : ℕ), n ≥ N → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N => @?_mvar.12521 NhN':∀ (N : ℕ), 2 * 10 ^ (-↑N - 1) ≤ 2 / (↑N + 1) := fun N => Trans.trans (@?_mvar.20396 N) (@?_mvar.20397 N)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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))hab':∀ (N n : ℕ), n ≥ N → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N => @?_mvar.12521 NhN':∀ (N : ℕ), 2 * 10 ^ (-↑N - 1) ≤ 2 / (↑N + 1) := fun N => Trans.trans (@?_mvar.20396 N) (@?_mvar.20397 N)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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))hab':∀ (N n : ℕ), n ≥ N → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N => @?_mvar.12521 NN: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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))hab':∀ (N n : ℕ), n ≥ N → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N => @?_mvar.12521 NN:ℕhN:2 * 10 ^ (-↑N - 1) ≤ ε⊢ ∀ 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), |@_fvar.5075 n - @_fvar.5310 n| = 2 * 10 ^ (-↑n - 1) := fun n => Trans.trans (Trans.trans rfl (@?_mvar.6548 n)) (abs_of_nonneg (@?_mvar.6549 n))hab':∀ (N n : ℕ), n ≥ N → |@_fvar.5075 n - @_fvar.5310 n| ≤ 2 * 10 ^ (-↑N - 1) := fun N => @?_mvar.12521 NN: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