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

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

Lemma CVU_limits_open (fn : nat -> R -> R) (D : R -> Prop) : open D -> CVU_dom fn D -> (forall x n, D x -> ex_finite_lim (fn n) x) -> forall x, D x -> ex_finite_lim_seq (fun n => real (Lim (fn n) x)) /\ ex_finite_lim (fun y => real (Lim_seq (fun n => fn n y))) x /\ real (Lim_seq (fun n => real (Lim (fn n) x))) = real (Lim (fun y => real (Lim_seq (fun n => fn n y))) x).