Require Export TopologicalSpaces OrderTopology Completeness MetricSpaces.
Require Export Compactness Connectedness.
Definition RTop := OrderTopology Rle.
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).