(** Uniform convergence *)
Definition CVU (fn:nat -> R -> R) (f:R -> R) (x:R)
  (r:posreal) : Prop :=
  forall eps:R,
    0 < eps ->
    exists N : nat,
      (forall (n:nat) (y:R),
        (N <= n)%nat -> Boule x r y -> Rabs (f y - fn n y) < eps).(** Normal convergence *)
Definition CVN_r (fn:nat -> R -> R) (r:posreal) : Type :=
  { An:nat -> R &
    { l:R |
      Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n) l /\
      (forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n) } }.Definition CVN_R (fn:nat -> R -> R) : Type := forall r:posreal, CVN_r fn r.Definition SFL (fn:nat -> R -> R)
  (cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l })
  (y:R) : R := let (a,_) := cv y in a.

Lemma continuity_pt_finite_SF : forall (fn:nat -> R -> R) (N:nat) (x:R), (forall n:nat, (n <= N)%nat -> continuity_pt (fn n) x) -> continuity_pt (fun y:R => sum_f_R0 (fun k:nat => fn k y) N) x.