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 filterlim_le : forall {T F} {FF : ProperFilter' F} (f g : T -> R) (lf lg : Rbar), F (fun x => f x <= g x) -> filterlim f F (Rbar_locally lf) -> filterlim g F (Rbar_locally lg) -> Rbar_le lf lg.