Definition ex_lim_seq_cauchy (u : nat -> R) :=
  forall eps : posreal, exists N : nat, forall n m,
    (N <= n)%nat -> (N <= m)%nat -> Rabs (u n - u m) < eps.

Lemma ex_lim_seq_adj (u v : nat -> R) : (forall n, u n <= u (S n)) -> (forall n, v (S n) <= v n) -> is_lim_seq (fun n => v n - u n) 0 -> ex_finite_lim_seq u /\ ex_finite_lim_seq v /\ Lim_seq u = Lim_seq v.