## A Dense Subset of a Quotient Space of a Metric Space By the Set of Cauchy Sequences

Theorem

Suppose we have a metric spaceSupposeis a Cauchy sequence in

Defineas follows:where

is dense in the quotient spacewhereis the equivalence relation on the set of Cauchy sequences indefined by:

if

thenis dense in

Proof

Let

Thenis a Cauchy sequence in

Henceis the limit of the sequenceinHenceis dense in