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.

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