Theorem: Asymptotics.IsLittleO.sum_range
File: Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean

Proof steps:
state_before: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
⊢ (fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => ∑ i ∈ range n, g i
state_after: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
⊢ (fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => ∑ i ∈ range n, g i
tactic: have A : ∀ i, ‖g i‖ = g i := fun i => Real.norm_of_nonneg (hg i)
state_before: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
⊢ (fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => ∑ i ∈ range n, g i
state_after: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
⊢ (fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => ∑ i ∈ range n, g i
tactic: have B : ∀ n, ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i := fun n => by
    rwa [Real.norm_eq_abs, abs_sum_of_nonneg']
state_before: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
⊢ (fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => ∑ i ∈ range n, g i
state_after: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
⊢ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (x : ℕ) in atTop, ‖∑ i ∈ range x, f i‖ ≤ ε * ‖∑ i ∈ range x, g i‖
tactic: apply isLittleO_iff.2 fun ε εpos => _
state_before: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
⊢ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (x : ℕ) in atTop, ‖∑ i ∈ range x, f i‖ ≤ ε * ‖∑ i ∈ range x, g i‖
state_after: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
ε : ℝ
εpos : 0 < ε
⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i ∈ range x, f i‖ ≤ ε * ‖∑ i ∈ range x, g i‖
tactic: intro ε εpos
state_before: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
ε : ℝ
εpos : 0 < ε
⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i ∈ range x, f i‖ ≤ ε * ‖∑ i ∈ range x, g i‖
state_after: case intro
α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
ε : ℝ
εpos : 0 < ε
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b
⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i ∈ range x, f i‖ ≤ ε * ‖∑ i ∈ range x, g i‖
tactic: obtain ⟨N, hN⟩ : ∃ N : ℕ, ∀ b : ℕ, N ≤ b → ‖f b‖ ≤ ε / 2 * g b := by
    simpa only [A, eventually_atTop] using isLittleO_iff.mp h (half_pos εpos)
state_before: case intro
α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
ε : ℝ
εpos : 0 < ε
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b
⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i ∈ range x, f i‖ ≤ ε * ‖∑ i ∈ range x, g i‖
state_after: case intro
α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
ε : ℝ
εpos : 0 < ε
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b
this : (fun x => ∑ i ∈ range N, f i) =o[atTop] fun n => ∑ i ∈ range n, g i
⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i ∈ range x, f i‖ ≤ ε * ‖∑ i ∈ range x, g i‖
tactic: have : (fun _ : ℕ => ∑ i ∈ range N, f i) =o[atTop] fun n : ℕ => ∑ i ∈ range n, g i := by
    apply isLittleO_const_left.2
    exact Or.inr (h'g.congr fun n => (B n).symm)
state_before: case intro
α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
ε : ℝ
εpos : 0 < ε
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b
this : (fun x => ∑ i ∈ range N, f i) =o[atTop] fun n => ∑ i ∈ range n, g i
⊢ ∀ᶠ (x : ℕ) in atTop, ‖∑ i ∈ range x, f i‖ ≤ ε * ‖∑ i ∈ range x, g i‖
state_after: case h
α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
B : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i
ε : ℝ
εpos : 0 < ε
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b
this : (fun x => ∑ i ∈ range N, f i) =o[atTop] fun n => ∑ i ∈ range n, g i
n : ℕ
hn : ‖∑ i ∈ range N, f i‖ ≤ ε / 2 * ‖∑ i ∈ range n, g i‖
Nn : n ∈ Set.Ici N
⊢ ‖∑ i ∈ range n, f i‖ ≤ ε * ‖∑ i ∈ range n, g i‖
tactic: filter_upwards [isLittleO_iff.1 this (half_pos εpos), Ici_mem_atTop N] with n hn Nn