Let $z_1,...,z_n\in \mathbb{C}$ and $f: \mathbb{C}\setminus \{z_1,...,z_n\} \to \mathbb{C}$ be holomorphic. Furthermore, $\operatorname{ord}(f,z_i)$ is finite for all $i$ and $\lim\limits_{|z|\to\infty}f(z)=0$. prove that $f$ is a rational function.
Taking a look at the main parts of the Laurent series expansion in $z_i$ might be helpful, but I don't know how to complete the proof.