Require Export MetricSpaces.
Section Completeness.
Variable X:Type.
Variable d:X->X->R.
Hypothesis d_metric: metric d.
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.

(* Specifications: *)
Lemma convergent_sequence_is_cauchy:
forall (x:Net nat_DS (MetricTopology d d_metric))
(x0:point_set (MetricTopology d d_metric)),
net_limit x x0 -> cauchy x.
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.