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_incr_compare (u : nat -> R) (l : R) : is_lim_seq u l -> (forall n, (u n) <= (u (S n))) -> forall n, u n <= l.