Proof That if the Quotient Space of the Cauchy Sequences in a Metric Space is a Completion of a Metric Space, then The Quotient Space is Isomorphic to That Space

Theorem

Suppose we have a metric space

Definewithifand

then ifis a completion of some metric spacethenis ismorphic to

Proof

is a subspace of

Hence for everythere exists a Cauchy sequenceconverging to

Define a mappingby

The mappingis well defined, since ifconverges tothenso that

Also,is subjective. Supposethenis a Cauchy sequence inButis complete henceconverges toand

Now supposethen there are sequencesinsuch that

Then

Henceis an isometry betweenand