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_le_loc (u v : nat -> R) : eventually (fun n => u n <= v n) -> Rbar_le (Lim_seq u) (Lim_seq v).