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 is_lim_seq_abs_0 (u : nat -> R) : is_lim_seq u 0 <-> is_lim_seq (fun n => Rabs (u n)) 0.