A Cauchy Sequence That is Not Convergent
If a sequenceis convergent in a metric spaceso that asit must be a Cauchy sequence, so that giventhere existssuch that for
The converse is not the case, so a sequence may be Cauchy without being convergent.
Consider the spacewith the absolute value metric. The sequence is defined as follows:
Each termin the sequence toto n significant figures. This sequence converges to butsodoes not converge.