I am stuck on the following question.
Suppose $f = u+iv$ is entire and there exists $M > 0$ such that $|u(z)| \leq M$ for all $z\in C$. Show that $f$ is constant.
I would figure we would have to use Liouville's Theorem to show this is true because the theorem states if "A bounded entire function is constant."
How would I continue to approach this problem?
I want to simply say because the function is entire and $|u(z)|$ is bounded that $f$ is constant. The only thing that makes me uneasy is $v(z)$. We do not know if it is bounded or not. Hence why I am stuck.
Thank you for your time and Thanks in advanced for any feedback.