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_abs (u : nat -> R) : ex_lim_seq u -> Lim_seq (fun n => Rabs (u n)) = Rbar_abs (Lim_seq u).