Proof That the Quotient of the Set of Cauchy Sequences in a Metric Space is a Completion of the Metric Space

Theorem

Ifis the quotient of a metric spacedefined bywhereis the equivalence relation on the set of Cauchy sequences indefined by:

if

with

Thenis a completion of

Proof

Lebe a Cauchy sequence in

is dense in Proof here.

Hence for allthere existssuch that

Henceis also a Cauchy sequence, converging to Proof here.

Thenconverges to

The conclusion is thatis complete.