Closure in $H^1((0,L)\times\mathbb{R})$

Question

Let $\Omega=(0,L)\times\mathbb{R}$ and consider the space $$V:=\{f\in H^2(\Omega) \ \vert \ f'(0,y)=f'(L,y)=0 \ \forall \ y\in\mathbb{R}\}.$$

I would like to show that the closure of $V$ with respect to $H^1(\Omega)$-norm is equal to $H^1(\Omega)$.

Can one maybe argue that $C^\infty_c(\Omega)$ is contained in $V$ and since it is dense in $H^1(\Omega)$, then $\overline{V}=H^1(\Omega)$?

Any help is appreciated!