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_scal_l (u : nat -> R) (a : R) : Lim_seq (fun n => a * u n) = Rbar_mult a (Lim_seq u).