Theorem
Ifare Cauchy sequences in a metric spacesuch thatforthen

is also a Cauchy sequence

converges toif and only ifconverges to
Proof
For 1:
Applying the triangle inequality,
LetWe can findsuch that
Sinceis Cauchy, there existssuch that
Now letwe gethence is Cauchy.
For 2:
Using the triangle theorem again gives
Hence,but
Ifthen and
Similarly, ifthen