∀ {α : Type u_1} {K : Set α} {R : Type u_3} [inst : NormedCommRing R] [NormOneClass R] [CompleteSpace R]
  [inst_3 : TopologicalSpace α] [LocallyCompactSpace α] {f : ℕ → α → R},
  IsOpen K →
    ∀ {u : ℕ → ℝ},
      Summable u →
        (∀ᶠ (n : ℕ) in Filter.atTop, ∀ x ∈ K, ‖f n x‖ ≤ u n) →
          (∀ (n : ℕ), ContinuousOn (f n) K) →
            HasProdLocallyUniformlyOn (fun n x => 1 + f n x) (fun x => ∏' (i : ℕ), (1 + f i x)) K