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fix: refute Ex. 9.8.4 without StrictMono; label §6.3 examples #628

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fix: refute Ex. 9.8.4 without StrictMono; label §6.3 examples #628

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db78032
fix: refute inverse existence without strict monotonicity
Chessing234 Aug 3, 2026
c6da825
fix: disambiguate Examples 6.3.3–6.3.5 and Remark 6.3.7 labels
Chessing234 Aug 3, 2026
a123af9
ci: retrigger build (prior run hung after mathlib cache)
Chessing234 Aug 3, 2026
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18 changes: 9 additions & 9 deletions Analysis/Section_6_3.lean
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Expand Up @@ -26,22 +26,22 @@ noncomputable abbrev Sequence.sup (a:Sequence) : EReal := sSup { x | ∃ n ≥ a
/-- Definition 6.3.1 -/
noncomputable abbrev Sequence.inf (a:Sequence) : EReal := sInf { x | ∃ n ≥ a.m, x = a n }

/-- Example 6.3.3 -/
/-- Example 6.3.3 (a) -/
example : ((fun (n:ℕ) ↦ (-1:ℝ)^(n+1)):Sequence).sup = 1 := by sorry

/-- Example 6.3.3 -/
/-- Example 6.3.3 (b) -/
example : ((fun (n:ℕ) ↦ (-1:ℝ)^(n+1)):Sequence).inf = -1 := by sorry

/-- Example 6.3.4 / Exercise 6.3.1 -/
/-- Example 6.3.4 / Exercise 6.3.1 (a) -/
example : ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)):Sequence).sup = 1 := by sorry

/-- Example 6.3.4 / Exercise 6.3.1 -/
/-- Example 6.3.4 / Exercise 6.3.1 (b) -/
example : ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)):Sequence).inf = 0 := by sorry

/-- Example 6.3.5 -/
/-- Example 6.3.5 (a) -/
example : ((fun (n:ℕ) ↦ (n+1:ℝ)):Sequence).sup = ⊤ := by sorry

/-- Example 6.3.5 -/
/-- Example 6.3.5 (b) -/
example : ((fun (n:ℕ) ↦ (n+1:ℝ)):Sequence).inf = 1 := by sorry

abbrev Sequence.BddAboveBy (a:Sequence) (M:ℝ) : Prop := ∀ n ≥ a.m, a n ≤ M
Expand All @@ -68,13 +68,13 @@ theorem Sequence.sup_le_upper {a:Sequence} {M:EReal} (h: ∀ n ≥ a.m, a n ≤
theorem Sequence.exists_between_lt_sup {a:Sequence} {y:EReal} (h: y < a.sup ) :
∃ n ≥ a.m, y < a n ∧ a n ≤ a.sup := by sorry

/-- Remark 6.3.7 -/
/-- Remark 6.3.7 (a) -/
theorem Sequence.ge_inf {a:Sequence} {n:ℤ} (hn: n ≥ a.m) : a n ≥ a.inf := by sorry

/-- Remark 6.3.7 -/
/-- Remark 6.3.7 (b) -/
theorem Sequence.inf_ge_lower {a:Sequence} {M:EReal} (h: ∀ n ≥ a.m, a n ≥ M) : a.inf ≥ M := by sorry

/-- Remark 6.3.7 -/
/-- Remark 6.3.7 (c) -/
theorem Sequence.exists_between_gt_inf {a:Sequence} {y:EReal} (h: y > a.inf ) :
∃ n ≥ a.m, y > a n ∧ a n ≥ a.inf := by sorry

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12 changes: 10 additions & 2 deletions Analysis/Section_9_8.lean
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Expand Up @@ -154,8 +154,16 @@ def MonotoneOn.exist_inverse_without_strictmono :
finv '' (.Icc (f a) (f b)) = .Icc a b ∧
(∀ x ∈ Set.Icc a b, finv (f x) = x) ∧
∀ y ∈ Set.Icc (f a) (f b), f (finv y) = y) := by
-- apply isFalse: e.g. a constant monotone f on [a,b] has no strict inverse
sorry
-- Constant continuous monotone maps collapse [a,b] to a point, so no left inverse.
apply isFalse
intro h
let f : ℝ → ℝ := fun _ ↦ 0
obtain ⟨_, finv, _, _, _, hleft, _⟩ :=
h 0 1 (by norm_num) f continuousOn_const fun _ _ _ _ _ ↦ le_rfl
have h0 := hleft 0 (by simp)
have h1 := hleft 1 (by simp)
simp [f] at h0 h1
linarith


/-
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