Show that a double-periodic and entire function is a constant function

Question

I know that a double-periodic function is a function $f$ such that $f(z)=f(z+2\pi)$ and $f(z)=f(z+2\pi i)$. My idea is to use Louisville's Theorem which states that if $f$ is entire and bounded in the $\mathbb{C}$-plane, then $f$ is constant throughout the $\mathbb{C}$-plane. Since $f$ is entire, I know that I only have to show that $f$ is bounded. But does it simply follow that $f$ is periodic means $f$ is bounded?

Answer 1

If it's continous, consider that the values of $f$ are completely determined by its values on $R=[0,2\pi]\times[0,2\pi]\subseteq\Bbb R^2\cong \Bbb C$. But then $f(R)$ is bounded because it is the image of a compact set under a continuous map. So $f$ is bounded on $\Bbb C$.