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 Lim_seq_geom_p (q : R) : 1 < q -> Lim_seq (fun n => q ^ n) = p_infty.