= Solution
If $A$ is closed in the complete ambient <metric space>, any <Cauchy sequence> in $A$ converges in $X$ by completeness, and its limit belongs to $A$ by part (i). Thus $A$ is complete with its restricted metric.
Conversely, suppose $A$ is complete and a <sequence> $a_n\in A$ converges in $X$ to $x$. A convergent <sequence> is Cauchy, also in the restricted metric, so completeness of $A$ gives $a_n\to a$ for some $a\in A$. Uniqueness of a metric limit gives $x=a\in A$. Part (i) now makes $A$ closed. Hence
$$
\boxed{A\text{ complete}\quad\Longleftrightarrow\quad A\text{ closed in }X.}
$$
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