∀ {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] {g : β → α} {s : Set β} [inst_1 : UniformSpace α] {f : ℕ → β → α},
  HasProdUniformlyOn f g s → TendstoUniformlyOn (fun x1 x2 => ∏ i ∈ Finset.range x1, f i x2) g Filter.atTop s