Theorem: limsup_eq_tendsto_sum_indicator_atTop
File: Mathlib/Topology/Algebra/Order/LiminfLimsup.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
R✝ : Type u_4
S : Type u_5
π : ι → Type u_6
R : Type u_7
inst✝¹ : StrictOrderedSemiring R
inst✝ : Archimedean R
s : ℕ → Set α
⊢ limsup s atTop = {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop}
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
R✝ : Type u_4
S : Type u_5
π : ι → Type u_6
R : Type u_7
inst✝¹ : StrictOrderedSemiring R
inst✝ : Archimedean R
s : ℕ → Set α
⊢ {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop} =
    {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop}
tactic: rw [limsup_eq_tendsto_sum_indicator_nat_atTop s]
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
R✝ : Type u_4
S : Type u_5
π : ι → Type u_6
R : Type u_7
inst✝¹ : StrictOrderedSemiring R
inst✝ : Archimedean R
s : ℕ → Set α
⊢ {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop} =
    {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop}
state_after: case h
ι : Type u_1
α : Type u_2
β : Type u_3
R✝ : Type u_4
S : Type u_5
π : ι → Type u_6
R : Type u_7
inst✝¹ : StrictOrderedSemiring R
inst✝ : Archimedean R
s : ℕ → Set α
ω : α
⊢ ω ∈ {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop} ↔
    ω ∈ {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop}
tactic: ext ω
state_before: case h
ι : Type u_1
α : Type u_2
β : Type u_3
R✝ : Type u_4
S : Type u_5
π : ι → Type u_6
R : Type u_7
inst✝¹ : StrictOrderedSemiring R
inst✝ : Archimedean R
s : ℕ → Set α
ω : α
⊢ ω ∈ {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop} ↔
    ω ∈ {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop}
state_after: case h
ι : Type u_1
α : Type u_2
β : Type u_3
R✝ : Type u_4
S : Type u_5
π : ι → Type u_6
R : Type u_7
inst✝¹ : StrictOrderedSemiring R
inst✝ : Archimedean R
s : ℕ → Set α
ω : α
⊢ Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop ↔
    Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop
tactic: simp only [Set.mem_setOf_eq]
state_before: case h
ι : Type u_1
α : Type u_2
β : Type u_3
R✝ : Type u_4
S : Type u_5
π : ι → Type u_6
R : Type u_7
inst✝¹ : StrictOrderedSemiring R
inst✝ : Archimedean R
s : ℕ → Set α
ω : α
⊢ Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop ↔
    Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop
state_after: case h
ι : Type u_1
α : Type u_2
β : Type u_3
R✝ : Type u_4
S : Type u_5
π : ι → Type u_6
R : Type u_7
inst✝¹ : StrictOrderedSemiring R
inst✝ : Archimedean R
s : ℕ → Set α
ω : α
⊢ Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop ↔
    Tendsto (fun n => ↑(∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω)) atTop atTop

ι : Type u_1
α : Type u_2
β : Type u_3
R✝ : Type u_4
S : Type u_5
π : ι → Type u_6
R : Type u_7
inst✝¹ : StrictOrderedSemiring R
inst✝ : Archimedean R
s : ℕ → Set α
ω : α
⊢ (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) = fun n =>
    ↑(∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω)
tactic: rw [(_ : (fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → R) ω) = fun n ↦
    ↑(∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → ℕ) ω))]