If $V\subset L^{p}(\mathbb{R})$ is closed, then the distance from $f$ to $V$ is the distance from $f$ to some point in $V$

Question

This a qualifying exam question. Let $1<p<\infty$ and $V\subset L^p(\mathbb{R})$ be closed. Define

$$d(f,V) = \inf_{v\in V} \|f-v\|_p.$$

Prove there exists $v_0\in V$ such that $d(f,v)=\|f-v_0\|_p$. The hint is to consider a minimizing sequence $v_n$ and extract a weakly convergent subsequence.

Well, since $d(f,V)$ is the infimum of a set of numbers, can extract a sequence $v_n\in V$ such that $\|f-v_n\|_p$ converges to $d(f,V)$. However I am stuck at this point. Any help would be greatly appreciated.