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_mult_l : forall (a : R) (l : Rbar), filterlim (Rmult a) (Rbar_locally l) (Rbar_locally (Rbar_mult a l)).