apply Filter.Eventually.mp h
case intro.intro
β : Type u_3
α : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
f✝ : α → β
inst✝² : CompleteLinearOrder β
inst✝¹ : OrderClosedTopology β
ι : Type u_1
f : ι → α → β
s : Set α
a : α
inst✝ : TopologicalSpace ι
I : Set ι
ne_I : Set.Nonempty I
hI : IsCompact I
hf : ∀ i ∈ I, LowerSemicontinuousWithinAt (f i) s a
hf' : UpperSemicontinuousOn (fun i => f i a) I
t : β
i : ι
ht : t < f i a
hi : i ∈ I
hi' : f i a = ⨆ x ∈ I, f x a
h : ∀ᶠ (x' : α) in 𝓝[s] a, t < f i x' := hf i hi t ht
⊢ ∀ᶠ (x' : α) in 𝓝[s] a, t < ⨆ i ∈ I, f i x'
case intro.intro
β : Type u_3
α : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
f✝ : α → β
inst✝² : CompleteLinearOrder β
inst✝¹ : OrderClosedTopology β
ι : Type u_1
f : ι → α → β
s : Set α
a : α
inst✝ : TopologicalSpace ι
I : Set ι
ne_I : Set.Nonempty I
hI : IsCompact I
hf : ∀ i ∈ I, LowerSemicontinuousWithinAt (f i) s a
hf' : UpperSemicontinuousOn (fun i => f i a) I
t : β
i : ι
ht : t < f i a
hi : i ∈ I
hi' : f i a = ⨆ x ∈ I, f x a
h : ∀ᶠ (x' : α) in 𝓝[s] a, t < f i x' := hf i hi t ht
⊢ ∀ᶠ (x : α) in 𝓝[s] a, t < f i x → t < ⨆ i ∈ I, f i x