(** ** Limits, integrals and differentiability *)

Definition is_connected (D : R -> Prop) :=
  forall a b x, D a -> D b -> a <= x <= b -> D x.

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).