Here is the statement of the question I am trying to proof:
Suppose $\Omega$ is a bounded open set in the complex plane, $f$ is an analytic function in $\Omega$ and $$\limsup_{n \to \infty} |f(z_n)| \leq M$$ for every sequence $\{z_n\}$ in $\Omega$ which converges to a boundary point of $\Omega.$ Prove that $|f(z)| \leq M$ for all $z \in \Omega.$
Here is the statement and the proof of the Maximum Modulus Thm. version III:
Could anyone tell me how can we adjust the proof of Maximum Modulus Thm. version III to prove this question,please?