(** ** Dini's theorem *)

Lemma Dini (fn : nat -> R -> R) (a b : R) :
a < b -> CVS_dom fn (fun x => a <= x <= b)
-> (forall (n : nat) (x : R), a <= x <= b -> continuity_pt (fn n) x)
-> (forall (x : R), a <= x <= b -> continuity_pt (fun y => Lim_seq (fun n => fn n y)) x)
-> (forall (n : nat) (x y : R), a <= x -> x <= y -> y <= b -> fn n x <= fn n y)
-> CVU_dom fn (fun x => a <= x <= b).