-- Mathlib/Algebra/Order/CauSeq/BigOperators.lean
lemma of_abv_le (n : ℕ) (hm : ∀ m, n ≤ m → abv (f m) ≤ a m) :
    IsCauSeq abs (fun n ↦ ∑ i ∈ range n, a i) → IsCauSeq abv fun n ↦ ∑ i ∈ range n, f i

Proof state:
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
hg : IsCauSeq abs fun n => ∑ i ∈ range n, a i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
j : ℕ
ji : j ≥ n ⊔ i
hi₁ : |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
hi₂ : |(fun n => ∑ i ∈ range n, a i) (n ⊔ i) - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
⊢ abv ((fun n => ∑ i ∈ range n, f i) j - (fun n => ∑ i ∈ range n, f i) (n ⊔ i)) < ε

Tactic:
have sub_le :=
  abs_sub_le (∑ k ∈ range j, a k) (∑ k ∈ range i, a k) (∑ k ∈ range (max n i), a k)

Result:
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
hg : IsCauSeq abs fun n => ∑ i ∈ range n, a i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
j : ℕ
ji : j ≥ n ⊔ i
hi₁ : |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
hi₂ : |(fun n => ∑ i ∈ range n, a i) (n ⊔ i) - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
sub_le :
  |∑ k ∈ range j, a k - ∑ k ∈ range (n ⊔ i), a k| ≤
    |∑ k ∈ range j, a k - ∑ k ∈ range i, a k| + |∑ k ∈ range i, a k - ∑ k ∈ range (n ⊔ i), a k|
⊢ abv ((fun n => ∑ i ∈ range n, f i) j - (fun n => ∑ i ∈ range n, f i) (n ⊔ i)) < ε

Proof state:
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i j : ℕ
ji : j ≥ n ⊔ i
k : ℕ
hk : j = k + n ⊔ i
⊢ abv (∑ i ∈ range (k + n ⊔ i), f i - ∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (k + n ⊔ i), a k - ∑ k ∈ range (n ⊔ i), a k

Tactic:
clear hk ji j

Result:
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i k : ℕ
⊢ abv (∑ i ∈ range (k + n ⊔ i), f i - ∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (k + n ⊔ i), a k - ∑ k ∈ range (n ⊔ i), a k

Proof state:
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
hg : IsCauSeq abs fun n => ∑ i ∈ range n, a i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
⊢ ∀ j ≥ n ⊔ i, abv ((fun n => ∑ i ∈ range n, f i) j - (fun n => ∑ i ∈ range n, f i) (n ⊔ i)) < ε

Tactic:
intro j ji

Result:
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
hg : IsCauSeq abs fun n => ∑ i ∈ range n, a i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2
j : ℕ
ji : j ≥ n ⊔ i
⊢ abv ((fun n => ∑ i ∈ range n, f i) j - (fun n => ∑ i ∈ range n, f i) (n ⊔ i)) < ε

Proof state:
case intro.succ
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i k' : ℕ
hi :
  abv (∑ i ∈ range (k' + n ⊔ i), f i - ∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (k' + n ⊔ i), a k - ∑ k ∈ range (n ⊔ i), a k
⊢ abv (f (k' + n ⊔ i) + (∑ x ∈ range (k' + n ⊔ i), f x + -∑ x ∈ range (n ⊔ i), f x)) ≤
    a (k' + n ⊔ i) + (∑ x ∈ range (k' + n ⊔ i), a x + -∑ x ∈ range (n ⊔ i), a x)

Tactic:
refine le_trans (abv_add _ _ _) ?_

Result:
case intro.succ
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i k' : ℕ
hi :
  abv (∑ i ∈ range (k' + n ⊔ i), f i - ∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (k' + n ⊔ i), a k - ∑ k ∈ range (n ⊔ i), a k
⊢ abv (f (k' + n ⊔ i)) + abv (∑ x ∈ range (k' + n ⊔ i), f x + -∑ x ∈ range (n ⊔ i), f x) ≤
    a (k' + n ⊔ i) + (∑ x ∈ range (k' + n ⊔ i), a x + -∑ x ∈ range (n ⊔ i), a x)

Proof state:
case intro
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i k : ℕ
⊢ abv (∑ i ∈ range (k + n ⊔ i), f i - ∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (k + n ⊔ i), a k - ∑ k ∈ range (n ⊔ i), a k

Tactic:
induction' k with k' hi

Result:
case intro.zero
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i : ℕ
⊢ abv (∑ i ∈ range (0 + n ⊔ i), f i - ∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (0 + n ⊔ i), a k - ∑ k ∈ range (n ⊔ i), a k

case intro.succ
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i k' : ℕ
hi :
  abv (∑ i ∈ range (k' + n ⊔ i), f i - ∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (k' + n ⊔ i), a k - ∑ k ∈ range (n ⊔ i), a k
⊢ abv (∑ i ∈ range (k' + 1 + n ⊔ i), f i - ∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (k' + 1 + n ⊔ i), a k - ∑ k ∈ range (n ⊔ i), a k

Proof state:
case intro.succ
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i k' : ℕ
hi :
  abv (∑ i ∈ range (k' + n ⊔ i), f i - ∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (k' + n ⊔ i), a k - ∑ k ∈ range (n ⊔ i), a k
⊢ abv (f (k' + n ⊔ i)) + abv (∑ x ∈ range (k' + n ⊔ i), f x + -∑ x ∈ range (n ⊔ i), f x) ≤
    a (k' + n ⊔ i) + (∑ x ∈ range (k' + n ⊔ i), a x + -∑ x ∈ range (n ⊔ i), a x)

Tactic:
simp only [sub_eq_add_neg] at hi

Result:
case intro.succ
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
a : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m
i k' : ℕ
hi :
  abv (∑ i ∈ range (k' + n ⊔ i), f i + -∑ i ∈ range (n ⊔ i), f i) ≤
    ∑ k ∈ range (k' + n ⊔ i), a k + -∑ k ∈ range (n ⊔ i), a k
⊢ abv (f (k' + n ⊔ i)) + abv (∑ x ∈ range (k' + n ⊔ i), f x + -∑ x ∈ range (n ⊔ i), f x) ≤
    a (k' + n ⊔ i) + (∑ x ∈ range (k' + n ⊔ i), a x + -∑ x ∈ range (n ⊔ i), a x)