apply LowerSemicontinuousOn.exists_iInf_of_isCompact (β := βᵒᵈ) ne_s hs hf
β : Type u_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
f : α → β
inst✝¹ : CompleteLinearOrder β
inst✝ : OrderClosedTopology β
s : Set α
ne_s : Set.Nonempty s
hs : IsCompact s
hf : UpperSemicontinuousOn f s
⊢ ∃ a ∈ s, f a = ⨆ x ∈ s, f x
no goals