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