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Feature/exercise #9

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3c4ce33
update 5_2
WhisperingDreams87 Aug 21, 2026
957a288
update 5_2
WhisperingDreams87 Aug 21, 2026
98899d9
update 5_2
WhisperingDreams87 Aug 21, 2026
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16 changes: 12 additions & 4 deletions Analysis/Section_5_2.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -221,24 +221,32 @@ lemma Sequence.equiv_example :
_ = |2 * (10:ℚ)^(-(n:ℤ)-1)| := by ring_nf
_ = _ := abs_of_nonneg (by positivity)
have hab' (N:ℕ) : ∀ n ≥ N, |a n - b n| ≤ 2 * 10 ^(-(N:ℤ)-1) := by
intro n hn; rw [hab n]; gcongr; norm_num
intro n hn;
rw [hab n];
gcongr;
norm_num
have hN : ∃ N:ℕ, 2 * (10:ℚ) ^(-(N:ℤ)-1) ≤ ε := by
have hN' (N:ℕ) : 2 * (10:ℚ)^(-(N:ℤ)-1) ≤ 2/(N+1) := calc
_ = 2 / (10:ℚ)^(N+1) := by
field_simp
simp [←Section_4_3.pow_eq_zpow, ←zpow_add₀ (show 10 ≠ (0:ℚ) by norm_num)]
simp [←Section_4_3.pow_eq_zpow]
simp [←zpow_add₀ (show 10 ≠ (0:ℚ) by norm_num)]
_ ≤ _ := by
gcongr
apply le_trans _ (pow_le_pow_left₀ (show 0 ≤ (2:ℚ) by norm_num)
(show (2:ℚ) ≤ 10 by norm_num) _)
convert Nat.cast_le.mpr (Section_4_3.two_pow_geq (N+1)) using 1 <;> try infer_instance
convert Nat.cast_le.mpr (Section_4_3.two_pow_geq (N+1))
using 1 <;>
try infer_instance
all_goals simp
choose N hN using exists_nat_gt (2 / ε)
refine ⟨ N, (hN' N).trans ?_ ⟩
rw [div_le_iff₀ (by positivity)]
rw [div_lt_iff₀ hε] at hN
grind [mul_comm]
choose N hN using hN; use N; intro n hn
choose N hN using hN;
use N;
intro n hn
linarith [hab' N n hn]

theorem Sequence.Equiv_Is_Symm {a b: ℕ → ℚ} (hab: Equiv a b) :
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