Definition cauchy (x:nat->X) : Prop :=
  forall eps:R, eps > 0 -> exists N:nat, forall m n:nat,
    (m >= N)%nat -> (n >= N)%nat -> d (x m) (x n) < eps.

Lemma cauchy_sequence_with_cluster_point_converges: forall (x:Net nat_DS (MetricTopology d d_metric)) (x0:point_set (MetricTopology d d_metric)), cauchy x -> net_cluster_point x x0 -> net_limit x x0.