Riesz lemma for $L^p$ space

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 $$1, then there exist $$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.

• This is an easy application of Hahn-Banach (hint: Every element of the dual of $L^p$ for $1<p<\infty$ attains its norm...) – David C. Ullrich Jan 12 at 13:06
• Can you give me further explanation how to apply that? – Hana Jan 16 at 11:08