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 ex_finite_lim_seq_decr (u : nat -> R) (M : R) : (forall n, (u (S n)) <= (u n)) -> (forall n, M <= u n) -> ex_finite_lim_seq u.