Let $f$ be a non-constant holomorphic function in a bounded open connected set $\Omega$ in $\mathbb{C}$. Let $M:=\lim \sup_{n \to \infty} |f(z_n)|$ for every sequence ${z_n}$ in $\Omega$ which converges to a boundary point of $\Omega$. Show that $|f(z)|<M$ for $z\in \Omega$.
I think this problem is another version of Maximum principle. But I can't catch the hint to solve. Could you give me some hint?