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_Rbar_inv : forall l : Rbar, l <> 0 -> filterlim Rinv (Rbar_locally l) (Rbar_locally (Rbar_inv l)).