Lean4
theorem _root_.cauchy_product (ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n))
(hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε :=
by
let ⟨P, hP⟩ := ha.bounded
let ⟨Q, hQ⟩ := hb.bounded
have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by simp) hP0)
let ⟨N, hN⟩ := hb.cauchy₂ hPε0
have hQε0 : 0 < ε / (4 * Q) :=
div_pos ε0 (mul_pos (show (0 : α) < 4 by simp) (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0)))
let ⟨M, hM⟩ := ha.cauchy₂ hQε0
refine ⟨2 * (max N M + 1), fun K hK ↦ ?_⟩
have h₁ : (∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k)) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n := by
simpa using sum_range_diag_flip K fun m n ↦ f m * g n
have h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k := by
simp [Finset.mul_sum]
have h₃ :
∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k :=
by rw [← sum_add_distrib]; simp [(mul_add _ _ _).symm]
have two_mul_two : (4 : α) = 2 * 2 := by norm_num
have hQ0 : Q ≠ 0 := fun h ↦ by simp [h] at hQε0
have h2Q0 : 2 * Q ≠ 0 := mul_ne_zero two_ne_zero hQ0
have hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε := by
rw [← div_div, div_mul_cancel₀ _ (Ne.symm (ne_of_lt hP0)), two_mul_two, mul_assoc, ← div_div,
div_mul_cancel₀ _ h2Q0, add_halves]
have hNMK : max N M + 1 < K := lt_of_lt_of_le (by rw [two_mul]; exact lt_add_of_pos_left _ (Nat.succ_pos _)) hK
have hKN : N < K :=
calc
N ≤ max N M := le_max_left _ _
_ < max N M + 1 := (Nat.lt_succ_self _)
_ < K := hNMK
have hsumlesum :
(∑ i ∈ range (max N M + 1), abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) ≤
∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P)) :=
by
gcongr with m hmJ
refine le_of_lt <| hN (K - m) (le_tsub_of_add_le_left <| hK.trans' ?_) K hKN.le
rw [two_mul]
gcongr
· exact (mem_range.1 hmJ).le
· exact Nat.le_succ_of_le (le_max_left _ _)
have hsumltP : (∑ n ∈ range (max N M + 1), abv (f n)) < P :=
calc
(∑ n ∈ range (max N M + 1), abv (f n)) = |∑ n ∈ range (max N M + 1), abv (f n)| :=
Eq.symm (abs_of_nonneg (sum_nonneg fun x _ ↦ abv_nonneg abv (f x)))
_ < P := hP (max N M + 1)
rw [h₁, h₂, h₃, sum_mul, ← sub_sub, sub_right_comm, sub_self, zero_sub, abv_neg abv]
refine lt_of_le_of_lt (IsAbsoluteValue.abv_sum _ _ _) ?_
suffices
(∑ i ∈ range (max N M + 1), abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) +
((∑ i ∈ range K, abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) -
∑ i ∈ range (max N M + 1), abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) <
ε / (2 * P) * P + ε / (4 * Q) * (2 * Q)
by
rw [hε] at this
simpa [abv_mul abv] using this
gcongr
· exact lt_of_le_of_lt hsumlesum (by rw [← sum_mul, mul_comm]; gcongr)
rw [sum_range_sub_sum_range (le_of_lt hNMK)]
calc
(∑ i ∈ range K with max N M + 1 ≤ i, abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) ≤
∑ i ∈ range K with max N M + 1 ≤ i, abv (f i) * (2 * Q) :=
by
gcongr
rw [sub_eq_add_neg]
grw [abv_add abv]
rw [two_mul, abv_neg abv]
gcongr <;> exact le_of_lt (hQ _)
_ < ε / (4 * Q) * (2 * Q) := by
rw [← sum_mul, ← sum_range_sub_sum_range (le_of_lt hNMK)]
have := lt_of_le_of_lt (abv_nonneg _ _) (hQ 0)
gcongr
exact
(le_abs_self _).trans_lt <|
hM _ ((Nat.le_succ_of_le (le_max_right _ _)).trans hNMK.le) _ <| Nat.le_succ_of_le <| le_max_right _ _
