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

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
⊢ 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: ext ω
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: 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
⊢ 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: 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))]
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: exact tendsto_natCast_atTop_iff.symm
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: no goals

tactic: ext n
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 : 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)
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
n : ℕ
⊢ ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω = ↑(∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω)

tactic: simp only [Set.indicator, Pi.one_apply, Finset.sum_boole, Nat.cast_id]
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
n : ℕ
⊢ ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω = ↑(∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω)
state_after: no goals