Theorem: Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet
File: Mathlib/Analysis/Complex/AbelLimit.lean

tactic: cases' le_or_lt M 1 with hM hM
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)
state_after: case inl
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : M ≤ 1
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)

case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)

tactic: let s := fun n ↦ ∑ i ∈ range n, f i
state_before: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)
state_after: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)

tactic: let g := fun z ↦ ∑' n, f n * z ^ n
state_before: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)
state_after: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)

tactic: have hm := Metric.tendsto_atTop.mp h
state_before: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)
state_after: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (∑ i ∈ range n, f i) l < ε
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)

tactic: rw [Metric.tendsto_nhdsWithin_nhds]
state_before: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (∑ i ∈ range n, f i) l < ε
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)
state_after: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (∑ i ∈ range n, f i) l < ε
⊢ ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → dist x 1 < δ → dist (∑' (n : ℕ), f n * x ^ n) l < ε

tactic: simp only [dist_eq_norm] at hm ⊢
state_before: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (∑ i ∈ range n, f i) l < ε
⊢ ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → dist x 1 < δ → dist (∑' (n : ℕ), f n * x ^ n) l < ε
state_after: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
⊢ ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε

tactic: intro ε εpos
state_before: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
⊢ ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε
state_after: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
⊢ ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε

tactic: obtain ⟨B₁, hB₁⟩ := hm (ε / 4 / M) (by positivity)
state_before: case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
⊢ ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε
state_after: case inr.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
⊢ ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε

tactic: let F := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
state_before: case inr.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
⊢ ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε
state_after: case inr.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
⊢ ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε

tactic: use ε / 4 / (F + 1), by positivity
state_before: case inr.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
⊢ ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε
state_after: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
⊢ ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < ε / 4 / (F + 1) → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε

tactic: intro z ⟨zn, zm⟩ zd
state_before: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
⊢ ∀ {x : ℂ}, x ∈ stolzSet M → ‖x - 1‖ < ε / 4 / (F + 1) → ‖∑' (n : ℕ), f n * x ^ n - l‖ < ε
state_after: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε

tactic: have p := abel_aux h zn
state_before: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε
state_after: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
p :
  Tendsto (fun n => (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i) atTop
    (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε

tactic: simp_rw [Metric.tendsto_atTop, dist_eq_norm, norm_sub_rev] at p
state_before: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
p :
  Tendsto (fun n => (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i) atTop
    (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε
state_after: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
p :
  ∀ ε > 0,
    ∃ N, ∀ n ≥ N, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε

tactic: obtain ⟨B₂, hB₂⟩ := p (ε / 2) (by positivity)
state_before: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
p :
  ∀ ε > 0,
    ∃ N, ∀ n ≥ N, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε
state_after: case right.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
p :
  ∀ ε > 0,
    ∃ N, ∀ n ≥ N, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε
B₂ : ℕ
hB₂ : ∀ n ≥ B₂, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε

tactic: clear hm p
state_before: case right.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
p :
  ∀ ε > 0,
    ∃ N, ∀ n ≥ N, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε
B₂ : ℕ
hB₂ : ∀ n ≥ B₂, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε
state_after: case right.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ∀ n ≥ B₂, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε

tactic: replace hB₂ := hB₂ (max B₁ B₂) (by simp)
state_before: case right.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ∀ n ≥ B₂, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε
state_after: case right.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε

tactic: suffices ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2 by
  calc
    _ = ‖l - g z‖ := by rw [norm_sub_rev]
    _ = ‖l - g z - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i +
        (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := by rw [sub_add_cancel _]
    _ ≤ ‖l - g z - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ +
        ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := norm_add_le _ _
    _ < ε / 2 + ε / 2 := add_lt_add hB₂ this
    _ = _ := add_halves ε
state_before: case right.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε
state_after: case right.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2

tactic: calc
  _ = ‖(1 - z) * ∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i +
      (1 - z) * ∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := by
    rw [← mul_add, sum_range_add_sum_Ico _ (le_max_left B₁ B₂)]
  _ ≤ ‖(1 - z) * ∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i‖ +
      ‖(1 - z) * ∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := norm_add_le _ _
  _ = ‖1 - z‖ * ‖∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i‖ +
      ‖1 - z‖ * ‖∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := by
    rw [norm_mul, norm_mul]
  _ ≤ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i +
      ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i := by
    gcongr <;> simp_rw [← norm_pow, ← norm_mul, norm_sum_le]
state_before: case right.intro
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2
state_after: case right.intro.calc.step
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i + ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i <
    ε / 2

tactic: have S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4 :=
  calc
    _ ≤ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ := by
      gcongr; nth_rw 3 [← mul_one ‖_‖]
      gcongr; exact pow_le_one _ (norm_nonneg _) zn.le
    _ ≤ ‖1 - z‖ * (F + 1) := by gcongr; linarith only
    _ < _ := by rwa [norm_sub_rev, lt_div_iff (by positivity)] at zd
state_before: case right.intro.calc.step
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i + ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i <
    ε / 2
state_after: case right.intro.calc.step
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i + ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i <
    ε / 2

tactic: have S₂ : ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4 :=
  calc
    _ ≤ ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ε / 4 / M * ‖z‖ ^ i := by
      gcongr with i hi
      have := hB₁ (i + 1) (by linarith only [(mem_Ico.mp hi).1])
      rw [norm_sub_rev] at this
      exact this.le
    _ = ‖1 - z‖ * (ε / 4 / M) * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖z‖ ^ i := by
      rw [← mul_sum, ← mul_assoc]
    _ ≤ ‖1 - z‖ * (ε / 4 / M) * ∑' i, ‖z‖ ^ i := by
      gcongr
      exact sum_le_tsum _ (fun _ _ ↦ by positivity)
        (summable_geometric_of_lt_one (by positivity) zn)
    _ = ‖1 - z‖ * (ε / 4 / M) / (1 - ‖z‖) := by
      rw [tsum_geometric_of_lt_one (by positivity) zn, ← div_eq_mul_inv]
    _ < M * (1 - ‖z‖) * (ε / 4 / M) / (1 - ‖z‖) := by gcongr; linarith only [zn]
    _ = _ := by
      rw [← mul_rotate, mul_div_cancel_right₀ _ (by linarith only [zn]),
        div_mul_cancel₀ _ (by linarith only [hM])]
state_before: case right.intro.calc.step
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i + ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i <
    ε / 2
state_after: case right.intro.calc.step
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
S₂ : ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i + ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i <
    ε / 2

tactic: convert add_lt_add S₁ S₂ using 1
state_before: case right.intro.calc.step
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
S₂ : ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i + ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i <
    ε / 2
state_after: case h.e'_4
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
S₂ : ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ε / 2 = ε / 4 + ε / 4

tactic: linarith only
state_before: case h.e'_4
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
S₂ : ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ε / 2 = ε / 4 + ε / 4
state_after: no goals

tactic: simp_rw [stolzSet_empty hM, nhdsWithin_empty, tendsto_bot]
state_before: case inl
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : M ≤ 1
⊢ Tendsto (fun z => ∑' (n : ℕ), f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)
state_after: no goals

tactic: positivity
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
⊢ ε / 4 / M > 0
state_after: no goals

tactic: positivity
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
⊢ ε / 4 / (F + 1) > 0
state_after: no goals

tactic: positivity
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
p :
  ∀ ε > 0,
    ∃ N, ∀ n ≥ N, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε
⊢ ε / 2 > 0
state_after: no goals

tactic: simp
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ∀ n ≥ B₂, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ max B₁ B₂ ≥ B₂
state_after: no goals

tactic: calc
  _ = ‖l - g z‖ := by rw [norm_sub_rev]
  _ = ‖l - g z - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i +
      (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := by rw [sub_add_cancel _]
  _ ≤ ‖l - g z - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ +
      ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := norm_add_le _ _
  _ < ε / 2 + ε / 2 := add_lt_add hB₂ this
  _ = _ := add_halves ε
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
this : ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε
state_after: no goals

tactic: rw [norm_sub_rev]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
this : ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ = ‖l - g z‖
state_after: no goals

tactic: rw [sub_add_cancel _]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
this : ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2
⊢ ‖l - g z‖ =
    ‖l - g z - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i +
        (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖
state_after: no goals

tactic: rw [← mul_add, sum_range_add_sum_Ico _ (le_max_left B₁ B₂)]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ =
    ‖(1 - z) * ∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i + (1 - z) * ∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖
state_after: no goals

tactic: rw [norm_mul, norm_mul]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖(1 - z) * ∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i‖ + ‖(1 - z) * ∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ =
    ‖1 - z‖ * ‖∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i‖ + ‖1 - z‖ * ‖∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖
state_after: no goals

tactic: gcongr <;> simp_rw [← norm_pow, ← norm_mul, norm_sum_le]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖1 - z‖ * ‖∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i‖ + ‖1 - z‖ * ‖∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ ≤
    ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i + ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i
state_after: no goals

tactic: gcongr
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i ≤ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖
state_after: case h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
i✝ : ℕ
a✝ : i✝ ∈ range B₁
⊢ ‖l - s (i✝ + 1)‖ * ‖z‖ ^ i✝ ≤ ‖l - s (i✝ + 1)‖

tactic: nth_rw 3 [← mul_one ‖_‖]
state_before: case h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
i✝ : ℕ
a✝ : i✝ ∈ range B₁
⊢ ‖l - s (i✝ + 1)‖ * ‖z‖ ^ i✝ ≤ ‖l - s (i✝ + 1)‖
state_after: case h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
i✝ : ℕ
a✝ : i✝ ∈ range B₁
⊢ ‖l - s (i✝ + 1)‖ * ‖z‖ ^ i✝ ≤ ‖l - s (i✝ + 1)‖ * 1

tactic: gcongr
state_before: case h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
i✝ : ℕ
a✝ : i✝ ∈ range B₁
⊢ ‖l - s (i✝ + 1)‖ * ‖z‖ ^ i✝ ≤ ‖l - s (i✝ + 1)‖ * 1
state_after: case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
i✝ : ℕ
a✝ : i✝ ∈ range B₁
⊢ ‖z‖ ^ i✝ ≤ 1

tactic: exact pow_le_one _ (norm_nonneg _) zn.le
state_before: case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
i✝ : ℕ
a✝ : i✝ ∈ range B₁
⊢ ‖z‖ ^ i✝ ≤ 1
state_after: no goals

tactic: gcongr
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ ≤ ‖1 - z‖ * (F + 1)
state_after: case h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ∑ i ∈ range B₁, ‖l - s (i + 1)‖ ≤ F + 1

tactic: linarith only
state_before: case h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ∑ i ∈ range B₁, ‖l - s (i + 1)‖ ≤ F + 1
state_after: no goals

tactic: rwa [norm_sub_rev, lt_div_iff (by positivity)] at zd
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖1 - z‖ * (F + 1) < ε / 4
state_after: no goals

tactic: positivity
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖1 - z‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ 0 < F + 1
state_after: no goals

tactic: gcongr with i hi
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i ≤
    ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ε / 4 / M * ‖z‖ ^ i
state_after: case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
⊢ ‖l - s (i + 1)‖ ≤ ε / 4 / M

tactic: have := hB₁ (i + 1) (by linarith only [(mem_Ico.mp hi).1])
state_before: case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
⊢ ‖l - s (i + 1)‖ ≤ ε / 4 / M
state_after: case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
this : ‖∑ i ∈ range (i + 1), f i - l‖ < ε / 4 / M
⊢ ‖l - s (i + 1)‖ ≤ ε / 4 / M

tactic: rw [norm_sub_rev] at this
state_before: case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
this : ‖∑ i ∈ range (i + 1), f i - l‖ < ε / 4 / M
⊢ ‖l - s (i + 1)‖ ≤ ε / 4 / M
state_after: case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
this : ‖l - ∑ i ∈ range (i + 1), f i‖ < ε / 4 / M
⊢ ‖l - s (i + 1)‖ ≤ ε / 4 / M

tactic: exact this.le
state_before: case h.h.h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
this : ‖l - ∑ i ∈ range (i + 1), f i‖ < ε / 4 / M
⊢ ‖l - s (i + 1)‖ ≤ ε / 4 / M
state_after: no goals

tactic: linarith only [(mem_Ico.mp hi).1]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
⊢ i + 1 ≥ B₁
state_after: no goals

tactic: rw [← mul_sum, ← mul_assoc]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ε / 4 / M * ‖z‖ ^ i = ‖1 - z‖ * (ε / 4 / M) * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖z‖ ^ i
state_after: no goals

tactic: gcongr
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * (ε / 4 / M) * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖z‖ ^ i ≤ ‖1 - z‖ * (ε / 4 / M) * ∑' (i : ℕ), ‖z‖ ^ i
state_after: case h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ∑ i ∈ Ico B₁ (max B₁ B₂), ‖z‖ ^ i ≤ ∑' (i : ℕ), ‖z‖ ^ i

tactic: exact sum_le_tsum _ (fun _ _ ↦ by positivity)
  (summable_geometric_of_lt_one (by positivity) zn)
state_before: case h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ∑ i ∈ Ico B₁ (max B₁ B₂), ‖z‖ ^ i ≤ ∑' (i : ℕ), ‖z‖ ^ i
state_after: no goals

tactic: positivity
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
x✝¹ : ℕ
x✝ : x✝¹ ∉ Ico B₁ (max B₁ B₂)
⊢ 0 ≤ ‖z‖ ^ x✝¹
state_after: no goals

tactic: positivity
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ 0 ≤ ‖z‖
state_after: no goals

tactic: rw [tsum_geometric_of_lt_one (by positivity) zn, ← div_eq_mul_inv]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * (ε / 4 / M) * ∑' (i : ℕ), ‖z‖ ^ i = ‖1 - z‖ * (ε / 4 / M) / (1 - ‖z‖)
state_after: no goals

tactic: gcongr
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * (ε / 4 / M) / (1 - ‖z‖) < M * (1 - ‖z‖) * (ε / 4 / M) / (1 - ‖z‖)
state_after: case hc
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ 0 < 1 - ‖z‖

tactic: linarith only [zn]
state_before: case hc
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ 0 < 1 - ‖z‖
state_after: no goals

tactic: rw [← mul_rotate, mul_div_cancel_right₀ _ (by linarith only [zn]),
  div_mul_cancel₀ _ (by linarith only [hM])]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ M * (1 - ‖z‖) * (ε / 4 / M) / (1 - ‖z‖) = ε / 4
state_after: no goals

tactic: linarith only [zn]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ 1 - ‖z‖ ≠ 0
state_after: no goals

tactic: linarith only [hM]
state_before: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ M ≠ 0
state_after: no goals