Show, that $ U = \left\{ u \in W_p^2 (\mathbb{R}_+)\ | \ u(0) = 0 \right\}$ is a closed subspace of $W_p^2 (\mathbb{R}_+)$

Question

I want to prove, that $U$ is a complete space.

I already proved, that $W_p^2 (\mathbb{R}_+)$ is complete.

But, as far as I've understood, you need to additionally show that $U$ is also a closed space, in order to claim that $U$ is complete, too.

But how should I do this? Because all the proofs of closeness I know would rely on knowing whether $U$ is complete or not (because I construct a sequence of elements from $U$, which I want to find out that it's limit belongs to $U$, too).

Any suggestions?