If $f$ is a entire function such that $f(z+n+im)=f(z)$ for all $z\in \mathbb{C}$ and for all $n,m \in \mathbb{Z}$, then $f$ is constant. If $f$ is a entire function such that $f(z+n+im)=f(z)$ for all $z\in \mathbb{C}$ and for all $n,m \in \mathbb{Z}$, then $f$ is constant.
I'm having trouble solving this one. Could you help me?
I have tried to get to that $f$ is bounded to apply the Liouville theorem, but I do not know how to limit this function, could someone help me please? Thank you.

Since $f$ is continuous, $f(R)$ is a compact subset of $\mathbb C$, where $R$ is the rectangle $\{a+bi\mid a, b\in[0, 1]\}$. Hence $f(R)$ is bounded. By that $f(z+n+im)=f(z)$, it follows that $f(\mathbb C)=f(R)$. Thus $f$ is a bounded entire function. Then the Liouville theorem entails that $f$ is constant.
