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 filterlim_Rbar_plus : forall x y z, is_Rbar_plus x y z -> filterlim (fun z => fst z + snd z) (filter_prod (Rbar_locally x) (Rbar_locally y)) (Rbar_locally z).