A Condition for Cauchy Sequences to Converge to the Same Limit


Ifare Cauchy sequences in a metric spacesuch thatforthen

  1. is also a Cauchy sequence

  2. converges toif and only ifconverges to


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


Ifthen and

Similarly, ifthen