refine isBigOWith_iff.mpr <| eventually_of_mem ?_ (fun x hx ↦ ?_) (U := s ×ˢ Set.univ)
α : Type u_1
ι : Type u_2
E : Type u_3
F : Type u_4
s : Set ι
inst✝² : TopologicalSpace ι
C : ι → E
c : F
inst✝¹ : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn C s
hs : IsCompact s
hC : ∀ i ∈ s, C i ≠ 0
l : Filter α
⊢ IsBigOWith (‖c‖ / sInf (norm '' (C '' s))) (𝓟 s ×ˢ l) (fun x => c) fun x =>
    match x with
    | (i, _x) => C i
case refine_1
α : Type u_1
ι : Type u_2
E : Type u_3
F : Type u_4
s : Set ι
inst✝² : TopologicalSpace ι
C : ι → E
c : F
inst✝¹ : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn C s
hs : IsCompact s
hC : ∀ i ∈ s, C i ≠ 0
l : Filter α
⊢ s ×ˢ Set.univ ∈ 𝓟 s ×ˢ l

case refine_2
α : Type u_1
ι : Type u_2
E : Type u_3
F : Type u_4
s : Set ι
inst✝² : TopologicalSpace ι
C : ι → E
c : F
inst✝¹ : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn C s
hs : IsCompact s
hC : ∀ i ∈ s, C i ≠ 0
l : Filter α
x : ι × α
hx : x ∈ s ×ˢ Set.univ
⊢ ‖c‖ ≤
    ‖c‖ / sInf (norm '' (C '' s)) *
      ‖match x with
        | (i, _x) => C i‖