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fix: disambiguate duplicate Verso labels across Chapter 6 #656

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teorth merged 1 commit into from
Aug 23, 2026
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fix: disambiguate duplicate Verso labels across Chapter 6 #656

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4 changes: 2 additions & 2 deletions Analysis/Section_6_3.lean
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Expand Up @@ -20,10 +20,10 @@ Main constructions and results of this section:

namespace Chapter6

/-- Definition 6.3.1 -/
/-- Definition 6.3.1 (supremum of a sequence) -/
noncomputable abbrev Sequence.sup (a:Sequence) : EReal := sSup { x | ∃ n ≥ a.m, x = a n }

/-- Definition 6.3.1 -/
/-- Definition 6.3.1 (infimum of a sequence) -/
noncomputable abbrev Sequence.inf (a:Sequence) : EReal := sInf { x | ∃ n ≥ a.m, x = a n }

/-- Example 6.3.3 (a) -/
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8 changes: 4 additions & 4 deletions Analysis/Section_6_4.lean
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Expand Up @@ -201,19 +201,19 @@ theorem Sequence.tendsTo_iff_eq_limsup_liminf {a:Sequence} (c:ℝ) :
a.TendsTo c ↔ a.liminf = c ∧ a.limsup = c := by
sorry

/-- Lemma 6.4.13 (Comparison principle) / Exercise 6.4.4 -/
/-- Lemma 6.4.13 (Comparison principle, sup) / Exercise 6.4.4 -/
theorem Sequence.sup_mono {a b:Sequence} (hm: a.m = b.m) (hab: ∀ n ≥ a.m, a n ≤ b n) :
a.sup ≤ b.sup := by sorry

/-- Lemma 6.4.13 (Comparison principle) / Exercise 6.4.4 -/
/-- Lemma 6.4.13 (Comparison principle, inf) / Exercise 6.4.4 -/
theorem Sequence.inf_mono {a b:Sequence} (hm: a.m = b.m) (hab: ∀ n ≥ a.m, a n ≤ b n) :
a.inf ≤ b.inf := by sorry

/-- Lemma 6.4.13 (Comparison principle) / Exercise 6.4.4 -/
/-- Lemma 6.4.13 (Comparison principle, limsup) / Exercise 6.4.4 -/
theorem Sequence.limsup_mono {a b:Sequence} (hm: a.m = b.m) (hab: ∀ n ≥ a.m, a n ≤ b n) :
a.limsup ≤ b.limsup := by sorry

/-- Lemma 6.4.13 (Comparison principle) / Exercise 6.4.4 -/
/-- Lemma 6.4.13 (Comparison principle, liminf) / Exercise 6.4.4 -/
theorem Sequence.liminf_mono {a b:Sequence} (hm: a.m = b.m) (hab: ∀ n ≥ a.m, a n ≤ b n) :
a.liminf ≤ b.liminf := by sorry

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6 changes: 3 additions & 3 deletions Analysis/Section_6_5.lean
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Expand Up @@ -70,15 +70,15 @@ theorem Sequence.lim_of_power_decay {k:ℕ} :
· simp
simp [lim_eq, ha', eq_zero_of_pow_eq_zero hlim]

/-- Lemma 6.5.2 / Exercise 6.5.2 -/
/-- Lemma 6.5.2 (ratio of absolute value less than one) / Exercise 6.5.2 -/
theorem Sequence.lim_of_geometric {x:ℝ} (hx: |x| < 1) : ((fun (n:ℕ) ↦ x^n):Sequence).TendsTo 0 := by
sorry

/-- Lemma 6.5.2 / Exercise 6.5.2 -/
/-- Lemma 6.5.2 (ratio one) / Exercise 6.5.2 -/
theorem Sequence.lim_of_geometric' {x:ℝ} (hx: x = 1) : ((fun (n:ℕ) ↦ x^n):Sequence).TendsTo 1 := by
sorry

/-- Lemma 6.5.2 / Exercise 6.5.2 -/
/-- Lemma 6.5.2 (ratio minus one or of absolute value greater than one) / Exercise 6.5.2 -/
theorem Sequence.lim_of_geometric'' {x:ℝ} (hx: x = -1 ∨ |x| > 1) :
((fun (n:ℕ) ↦ x^n):Sequence).Divergent := by
sorry
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4 changes: 2 additions & 2 deletions Analysis/Section_6_6.lean
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Expand Up @@ -35,10 +35,10 @@ example :
(fun n ↦ (10:ℝ)^(-(n:ℤ)-1)) := by
sorry

/-- Lemma 6.6.4 / Exercise 6.6.1 -/
/-- Lemma 6.6.4 (reflexivity) / Exercise 6.6.1 -/
theorem Sequence.subseq_self (a:ℕ → ℝ) : Sequence.subseq a a := by sorry

/-- Lemma 6.6.4 / Exercise 6.6.1 -/
/-- Lemma 6.6.4 (transitivity) / Exercise 6.6.1 -/
theorem Sequence.subseq_trans {a b c:ℕ → ℝ} (hab: Sequence.subseq a b) (hbc: Sequence.subseq b c) :
Sequence.subseq a c := by sorry

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4 changes: 2 additions & 2 deletions Analysis/Section_6_7.lean
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Expand Up @@ -175,11 +175,11 @@ theorem Real.ratPow_neg {x:ℝ} (hx: x > 0) (q:ℝ) : rpow x (-q) = 1 / rpow x q
theorem Real.ratPow_mono {x y:ℝ} (hx: x > 0) (hy: y > 0) {q:ℝ} (h: q > 0) : x > y ↔ rpow x q > rpow y q := by
sorry

/-- Proposition 6.7.3(e) / Exercise 6.7.1 -/
/-- Proposition 6.7.3(e) (base greater than one) / Exercise 6.7.1 -/
theorem Real.ratPow_mono_of_gt_one {x:ℝ} (hx: x > 1) {q r:ℝ} : rpow x q > rpow x r ↔ q > r := by
sorry

/-- Proposition 6.7.3(e) / Exercise 6.7.1 -/
/-- Proposition 6.7.3(e) (base less than one) / Exercise 6.7.1 -/
theorem Real.ratPow_mono_of_lt_one {x:ℝ} (hx0: 0 < x) (hx: x < 1) {q r:ℝ} : rpow x q > rpow x r ↔ q < r := by
sorry

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