= Solution
A <complete metric space> is one in which \b[every <Cauchy sequence> converges to a point of that space]. For an incomplete example, take $(0,1)$ with the usual <Euclidean metric> and $x_n=1/(n+1)$. It is <Cauchy> because it converges to zero in $\mathbb R$. It cannot converge inside $(0,1)$: any such convergence would also hold in $\mathbb R$, contradicting uniqueness of the limit, since zero is excluded. Thus the ambient metric and the underlying set both matter to <completeness>.
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