Proof That the Quotient of the Set of Cauchy Sequences in a Metric Space is a Completion of the Metric Space
Ifis the quotient of a metric spacedefined bywhereis the equivalence relation on the set of Cauchy sequences indefined by:
Thenis a completion of
Lebe a Cauchy sequence in
is dense in Proof here.
Hence for allthere existssuch that
Henceis also a Cauchy sequence, converging to Proof here.
The conclusion is thatis complete.