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_le_loc (u v : nat -> R) (l1 l2 : Rbar) : eventually (fun n => u n <= v n) -> is_lim_seq u l1 -> is_lim_seq v l2 -> Rbar_le l1 l2.