Riesz lemma for $L^p$ space

Question

I need a proof for special case of Riesz lemma (when $\varepsilon$ is 0):

If Y is a closed proper subspace of $L^p(\mu)$ for some $1<p<\infty$, then there exist $f\in L^p(\mu)$ such that $||f||=1$ and $||f-g||\geq 1$ for every $g\in Y$.

I know that Clarkson's inequality (uniform convexity) can be used.