Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$

Question

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$.

I've already proved that if $U$ is a closed subspace then $U = (U^{\bot})^{\bot}$.

I also know that $v \in \bar V$ if and only if there exist a sequence $(v_n)$ in $V$ such that $v_n \rightarrow v$ as $n \rightarrow \infty$.