Require Export TopologicalSpaces.
Require Export OrderTopology.
Require Export Reals.
Definition RTop := OrderTopology Rle.
Require Export MetricSpaces.
Definition R_metric (x y:R) : R := Rabs (y-x).

Lemma R_cauchy_sequence_lower_bound: forall x:nat->R,
cauchy R_metric x -> lower_bound (Im Full_set x).