Let $\displaystyle f(z)$ be a function which is analytic on and inside the circle $\displaystyle |z| = 1$ and which satisfies $\displaystyle |f(z)|\leq{M}$ for all $\displaystyle z$ on the circle $\displaystyle |z| = 1$. Prove that $\displaystyle |f(0)|\leq{M}$.

By Cauchy's inequality i managed to obtain

$\displaystyle |f'(0)|\leq{\frac{1!M}{1^1}=M}$.

How do i proceed to prove that $\displaystyle |f(0)|\leq{M}$? Any help/suggestion is welcome!