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 : SeminormedAddGroup E
inst✝ : Norm F
hf : ContinuousOn C s
hs : IsCompact s
hc : ‖c‖ ≠ 0
l : Filter α
⊢ IsBigOWith (sSup (norm '' (C '' s)) / ‖c‖) (𝓟 s ×ˢ l)
    (fun x =>
      match x with
      | (i, _x) => C i)
    fun x => c
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 : SeminormedAddGroup E
inst✝ : Norm F
hf : ContinuousOn C s
hs : IsCompact s
hc : ‖c‖ ≠ 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 : SeminormedAddGroup E
inst✝ : Norm F
hf : ContinuousOn C s
hs : IsCompact s
hc : ‖c‖ ≠ 0
l : Filter α
x : ι × α
hx : x ∈ s ×ˢ Set.univ
⊢ ‖match x with
      | (i, _x) => C i‖ ≤
    sSup (norm '' (C '' s)) / ‖c‖ * ‖c‖