(** ** 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).