Analysis
6.3. Suprema and Infima of sequences

Imports

import Mathlib.Tactic
import Analysis.Section_6_1
import Analysis.Section_6_2
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Analysis I, Section 6.3: Suprema and infima of 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:

Suprema and infima of sequences.

namespace Chapter6

Definition 6.3.1

noncomputable abbrev Sequence.sup (a:Sequence) : EReal := sSup { x | ∃ n ≥ a.m, x = a n }

Definition 6.3.1

noncomputable abbrev Sequence.inf (a:Sequence) : EReal := sInf { x | ∃ n ≥ a.m, x = a n }

              /-- Example 6.3.3 -/
declaration uses `sorry`example : ((fun (n:ℕ) ↦ (-1:ℝ)^(n+1)):Sequence).sup = 1 := by⊢ (↑fun n ↦ (-1) ^ (n + 1)).sup = 1 sorryAll goals completed! 🐙/-- Example 6.3.3 -/
declaration uses `sorry`example : ((fun (n:ℕ) ↦ (-1:ℝ)^(n+1)):Sequence).inf = -1 := by⊢ (↑fun n ↦ (-1) ^ (n + 1)).inf = -1 sorryAll goals completed! 🐙/-- Example 6.3.4 / Exercise 6.3.1 -/
declaration uses `sorry`example : ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)):Sequence).sup = 1 := by⊢ (↑fun n ↦ 1 / (↑n + 1)).sup = 1 sorryAll goals completed! 🐙/-- Example 6.3.4 / Exercise 6.3.1 -/
declaration uses `sorry`example : ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)):Sequence).inf = 0 := by⊢ (↑fun n ↦ 1 / (↑n + 1)).inf = 0 sorryAll goals completed! 🐙/-- Example 6.3.5 -/
declaration uses `sorry`example : ((fun (n:ℕ) ↦ (n+1:ℝ)):Sequence).sup = ⊤ := by⊢ (↑fun n ↦ ↑n + 1).sup = ⊤ sorryAll goals completed! 🐙/-- Example 6.3.5 -/
declaration uses `sorry`example : ((fun (n:ℕ) ↦ (n+1:ℝ)):Sequence).inf = 1 := by⊢ (↑fun n ↦ ↑n + 1).inf = 1 sorryAll goals completed! 🐙abbrev Sequence.BddAboveBy (a:Sequence) (M:ℝ) : Prop := ∀ n ≥ a.m, a n ≤ Mabbrev Sequence.BddAbove (a:Sequence) : Prop := ∃ M, a.BddAboveBy Mabbrev Sequence.BddBelowBy (a:Sequence) (M:ℝ) : Prop := ∀ n ≥ a.m, a n ≥ Mabbrev Sequence.BddBelow (a:Sequence) : Prop := ∃ M, a.BddBelowBy Mtheorem declaration uses `sorry`Sequence.bounded_iff (a:Sequence) : a.IsBounded ↔ a.BddAbove ∧ a.BddBelow := bya:Sequence⊢ a.IsBounded ↔ a.BddAbove ∧ a.BddBelow sorryAll goals completed! 🐙theorem declaration uses `sorry`Sequence.sup_of_bounded {a:Sequence} (h: a.IsBounded) : a.sup.IsFinite := bya:Sequenceh:a.IsBounded⊢ a.sup.IsFinite sorryAll goals completed! 🐙theorem declaration uses `sorry`Sequence.inf_of_bounded {a:Sequence} (h: a.IsBounded) : a.inf.IsFinite := bya:Sequenceh:a.IsBounded⊢ a.inf.IsFinite sorryAll goals completed! 🐙

Proposition 6.3.6 (Least upper bound property) / Exercise 6.3.2

theorem declaration uses `sorry`Sequence.le_sup {a:Sequence} {n:ℤ} (hn: n ≥ a.m) : a n ≤ a.sup := bya:Sequencen:ℤhn:n ≥ a.m⊢ ↑(a.seq n) ≤ a.sup sorryAll goals completed! 🐙

Proposition 6.3.6 (Least upper bound property) / Exercise 6.3.2

theorem declaration uses `sorry`Sequence.sup_le_upper {a:Sequence} {M:EReal} (h: ∀ n ≥ a.m, a n ≤ M) : a.sup ≤ M := bya:SequenceM:ERealh:∀ n ≥ a.m, ↑(a.seq n) ≤ M⊢ a.sup ≤ M sorryAll goals completed! 🐙

Proposition 6.3.6 (Least upper bound property) / Exercise 6.3.2

theorem declaration uses `sorry`Sequence.exists_between_lt_sup {a:Sequence} {y:EReal} (h: y < a.sup ) :
    ∃ n ≥ a.m, y < a n ∧ a n ≤ a.sup := bya:Sequencey:ERealh:y < a.sup⊢ ∃ n ≥ a.m, y < ↑(a.seq n) ∧ ↑(a.seq n) ≤ a.sup sorryAll goals completed! 🐙

Remark 6.3.7

theorem declaration uses `sorry`Sequence.ge_inf {a:Sequence} {n:ℤ} (hn: n ≥ a.m) : a n ≥ a.inf := bya:Sequencen:ℤhn:n ≥ a.m⊢ ↑(a.seq n) ≥ a.inf sorryAll goals completed! 🐙

Remark 6.3.7

theorem declaration uses `sorry`Sequence.inf_ge_lower {a:Sequence} {M:EReal} (h: ∀ n ≥ a.m, a n ≥ M) : a.inf ≥ M := bya:SequenceM:ERealh:∀ n ≥ a.m, ↑(a.seq n) ≥ M⊢ a.inf ≥ M sorryAll goals completed! 🐙

Remark 6.3.7

theorem declaration uses `sorry`Sequence.exists_between_gt_inf {a:Sequence} {y:EReal} (h: y > a.inf ) :
    ∃ n ≥ a.m, y > a n ∧ a n ≥ a.inf := bya:Sequencey:ERealh:y > a.inf⊢ ∃ n ≥ a.m, y > ↑(a.seq n) ∧ ↑(a.seq n) ≥ a.inf sorryAll goals completed! 🐙

abbrev Sequence.IsMonotone (a:Sequence) : Prop := ∀ n ≥ a.m, a (n+1) ≥ a nabbrev Sequence.IsAntitone (a:Sequence) : Prop := ∀ n ≥ a.m, a (n+1) ≤ a n

Proposition 6.3.8 / Exercise 6.3.3

theorem declaration uses `sorry`Sequence.convergent_of_monotone {a:Sequence} (hbound: a.BddAbove) (hmono: a.IsMonotone) :
    a.Convergent := bya:Sequencehbound:a.BddAbovehmono:a.IsMonotone⊢ a.Convergent sorryAll goals completed! 🐙

Proposition 6.3.8 / Exercise 6.3.3

theorem declaration uses `sorry`Sequence.lim_of_monotone {a:Sequence} (hbound: a.BddAbove) (hmono: a.IsMonotone) :
    lim a = a.sup := bya:Sequencehbound:a.BddAbovehmono:a.IsMonotone⊢ ↑(lim a) = a.sup sorryAll goals completed! 🐙

              theorem declaration uses `sorry`Sequence.convergent_of_antitone {a:Sequence} (hbound: a.BddBelow) (hmono: a.IsAntitone) :
    a.Convergent := bya:Sequencehbound:a.BddBelowhmono:a.IsAntitone⊢ a.Convergent sorryAll goals completed! 🐙theorem declaration uses `sorry`Sequence.lim_of_antitone {a:Sequence} (hbound: a.BddBelow) (hmono: a.IsAntitone) :
    lim a = a.inf := bya:Sequencehbound:a.BddBelowhmono:a.IsAntitone⊢ ↑(lim a) = a.inf sorryAll goals completed! 🐙theorem declaration uses `sorry`Sequence.convergent_iff_bounded_of_monotone {a:Sequence} (ha: a.IsMonotone) :
    a.Convergent ↔ a.IsBounded := bya:Sequenceha:a.IsMonotone⊢ a.Convergent ↔ a.IsBounded sorryAll goals completed! 🐙theorem declaration uses `sorry`Sequence.bounded_iff_convergent_of_antitone {a:Sequence} (ha: a.IsAntitone) :
    a.Convergent ↔ a.IsBounded := bya:Sequenceha:a.IsAntitone⊢ a.Convergent ↔ a.IsBounded sorryAll goals completed! 🐙

Example 6.3.9

noncomputable abbrev Example_6_3_9 (n:ℕ) := ⌊ Real.pi * 10^n ⌋ / (10:ℝ)^n

              /-- Example 6.3.9 -/
declaration uses `sorry`example : (Example_6_3_9:Sequence).IsMonotone := by⊢ (↑Example_6_3_9).IsMonotone sorryAll goals completed! 🐙/-- Example 6.3.9 -/
declaration uses `sorry`example : (Example_6_3_9:Sequence).BddAboveBy 4 := by⊢ (↑Example_6_3_9).BddAboveBy 4 sorryAll goals completed! 🐙/-- Example 6.3.9 -/
declaration uses `sorry`example : (Example_6_3_9:Sequence).Convergent := by⊢ (↑Example_6_3_9).Convergent sorryAll goals completed! 🐙/-- Example 6.3.9 -/
declaration uses `sorry`example : lim (Example_6_3_9:Sequence) ≤ 4 := by⊢ lim ↑Example_6_3_9 ≤ 4 sorryAll goals completed! 🐙

Proposition 6.3.1

theorem declaration uses `sorry`lim_of_exp {x:ℝ} (hpos: 0 < x) (hbound: x < 1) :
    ((fun (n:ℕ) ↦ x^n):Sequence).Convergent ∧ lim ((fun (n:ℕ) ↦ x^n):Sequence) = 0 := byx:ℝhpos:0 < xhbound:x < 1⊢ (↑fun n ↦ x ^ n).Convergent ∧ (lim ↑fun n ↦ x ^ n) = 0
  -- This proof is written to follow the structure of the original text.
  set a := ((fun (n:ℕ) ↦ x^n):Sequence)x:ℝhpos:0 < xhbound:x < 1a:Sequence := ↑fun n ↦ x ^ n⊢ a.Convergent ∧ lim a = 0
  have why : a.IsAntitone := sorryx:ℝhpos:0 < xhbound:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitone⊢ a.Convergent ∧ lim a = 0
  have hbound : a.BddBelowBy 0 := byx:ℝhpos:0 < xhbound:x < 1⊢ (↑fun n ↦ x ^ n).Convergent ∧ (lim ↑fun n ↦ x ^ n) = 0 intro n _x:ℝhpos:0 < xhbound:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonen:ℤa✝:n ≥ a.m⊢ a.seq n ≥ 0; positivityAll goals completed! 🐙
  have hbound' : a.BddBelow := byx:ℝhpos:0 < xhbound:x < 1⊢ (↑fun n ↦ x ^ n).Convergent ∧ (lim ↑fun n ↦ x ^ n) = 0 use 0All goals completed! 🐙
  have hconv := a.convergent_of_antitone hbound' whyx:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.Convergent⊢ a.Convergent ∧ lim a = 0
  set L := lim ax:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.ConvergentL:ℝ := lim a⊢ a.Convergent ∧ L = 0
  have : lim ((fun (n:ℕ) ↦ x^(n+1)):Sequence) = x * L := byx:ℝhpos:0 < xhbound:x < 1⊢ (↑fun n ↦ x ^ n).Convergent ∧ (lim ↑fun n ↦ x ^ n) = 0
    rw [←(a.lim_smul x hconv).2]x:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.ConvergentL:ℝ := lim a⊢ (lim ↑fun n ↦ x ^ (n + 1)) = lim (x • a); congre_ax:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.ConvergentL:ℝ := lim a⊢ (↑fun n ↦ x ^ (n + 1)) = x • a; ext ne_a.mx:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.ConvergentL:ℝ := lim a⊢ (↑fun n ↦ x ^ (n + 1)).m = (x • a).me_a.seq.hx:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.ConvergentL:ℝ := lim an:ℤ⊢ (↑fun n ↦ x ^ (n + 1)).seq n = (x • a).seq n; rfle_a.seq.hx:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.ConvergentL:ℝ := lim an:ℤ⊢ (↑fun n ↦ x ^ (n + 1)).seq n = (x • a).seq n
    simp [a, pow_succ', HSMul.hSMul, SMul.smul]All goals completed! 🐙
  have why2 : lim ((fun (n:ℕ) ↦ x^(n+1)):Sequence) = lim ((fun (n:ℕ) ↦ x^n):Sequence) := byx:ℝhpos:0 < xhbound:x < 1⊢ (↑fun n ↦ x ^ n).Convergent ∧ (lim ↑fun n ↦ x ^ n) = 0 sorryAll goals completed! 🐙
  convert_to x * L = 1 * L at why2h.e'_3x:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.ConvergentL:ℝ := lim athis:(lim ↑fun n ↦ x ^ (n + 1)) = x * Lwhy2:(lim ↑fun n ↦ x ^ (n + 1)) = lim ↑fun n ↦ x ^ n⊢ (lim ↑fun n ↦ x ^ n) = 1 * Lx:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.ConvergentL:ℝ := lim athis:(lim ↑fun n ↦ x ^ (n + 1)) = x * Lwhy2:x * L = 1 * L⊢ a.Convergent ∧ L = 0; simp [a,L]x:ℝhpos:0 < xhbound✝:x < 1a:Sequence := ↑fun n ↦ x ^ nwhy:a.IsAntitonehbound:a.BddBelowBy 0hbound':a.BddBelowhconv:a.ConvergentL:ℝ := lim athis:(lim ↑fun n ↦ x ^ (n + 1)) = x * Lwhy2:x * L = 1 * L⊢ a.Convergent ∧ L = 0
  have hx : x ≠ 1 := byx:ℝhpos:0 < xhbound:x < 1⊢ (↑fun n ↦ x ^ n).Convergent ∧ (lim ↑fun n ↦ x ^ n) = 0 grindAll goals completed! 🐙
  simp_all [-one_mul]All goals completed! 🐙

Exercise 6.3.4

theorem declaration uses `sorry`lim_of_exp' {x:ℝ} (hbound: x > 1) : ¬((fun (n:ℕ) ↦ x^n):Sequence).Convergent := byx:ℝhbound:x > 1⊢ ¬(↑fun n ↦ x ^ n).Convergent sorryAll goals completed! 🐙

end Chapter6