Consider a normed vector space $(V,\lVert \cdot \rVert)$. Need to show that if $S\subseteq V$ is complete then $S$ is closed.
A complete subset $S$ of $V$ satisfies that any sequence contained entirely in $S$ converges to a point in $S$, with respect to $\lVert \cdot \rVert$. Suppose $V$ is open. Then there exists a point $x\in V\backslash S$ such that $x$ is a limit point of $S$. But this contradicts the above definition. Hence, $V$ must be closed.
Please let me know if this approach is correct.