(** * Sequence of functions *)

(** ** Definitions *)

Definition CVS_dom (fn : nat -> R -> R) (D : R -> Prop) :=
  forall x : R, D x -> ex_finite_lim_seq (fun n => fn n x).Definition CVU_dom (fn : nat -> R -> R) (D : R -> Prop) :=
  forall eps : posreal, eventually (fun n => forall x : R,
    D x -> Rabs ((fn n x) - real (Lim_seq (fun n => fn n x))) < eps).Definition CVU_cauchy (fn : nat -> R -> R) (D : R -> Prop) :=
  forall eps : posreal, exists N : nat,
  forall (n m : nat) (x : R), D x -> (N <= n)%nat -> (N <= m)%nat
    -> Rabs (fn n x - fn m x) < eps.

Lemma CVU_dom_Reals (fn : nat -> R -> R) (f : R -> R) (x : R) (r : posreal) : (forall y, (Boule x r y) -> (Finite (f y)) = Lim_seq (fun n => fn n y)) -> (CVU fn f x r <-> CVU_dom fn (Boule x r)).