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 Lim_seq_plus (u v : nat -> R) : ex_lim_seq u -> ex_lim_seq v -> ex_Rbar_plus (Lim_seq u) (Lim_seq v) -> Lim_seq (fun n => u n + v n) = Rbar_plus (Lim_seq u) (Lim_seq v).