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 is_lim_seq_plus' (u v : nat -> R) (l1 l2 : R) : is_lim_seq u l1 -> is_lim_seq v l2 -> is_lim_seq (fun n => u n + v n) (l1 + l2).