If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

If $V$ is not complete and $V\times W$ complete, take $\{v_n\}$ a Cauchy sequence which doesn't converge in $V$. Then $(v_n,0)$ is a Cauchy sequence in $V\times W$ and converges to $(v,w)\in V\times W$. We have $$\lVert (v_n,0)-(v,w)\rVert_{V\times W}=\lVert v_n-v\rVert+\lVert w\rVert\to 0$$ hence $v_n\to v$ in $V$, a contradiction.

Hence $V\times W$ is complete if and only if so are $V$ and $W$.

I believe that the spaces $V$ and $W$ must be complete whenever $V\times W$ is complete.

Closed subspace of a complete normed space is complete.

The space $V$ is isometrically isomorphic to the closed subspace $V\times\{0\}$ of $V\times W$.