If a sequence
is convergent in a metric space
so that as
it must be a Cauchy sequence, so that given
there exists
such that for
The converse is not the case, so a sequence may be Cauchy without being convergent.
Consider the space
with the absolute value metric. The sequence is defined as follows:
Each term
in the sequence to
to n significant figures. This sequence converges to
but
sodoes not converge.