(** Uniform convergence *)
(** 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.

(** As R is complete, normal convergence implies that (fn) is simply-uniformly convergent *)
Lemma CVN_R_CVS :
forall fn:nat -> R -> R,
CVN_R fn -> forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }.