# meromorphic, rational function

Let $f(z)$ be a meromorphic function on the complex plane, and suppose there is an integer $m$ such that $f^{-1}(w)$ has at most $m$ points for all $w \in \mathbb{C}$. Show that $f(z)$ is a rational function.
Choose $w_0$ such that the number of points in $f^{-1}(w_0)$ is maximum. Then $f(z)$ attains values $w$ near $w_0$ only near points in $f^{-1}(w_0)$, $\frac{1}{f(z)-w_0}$ is bounded at $\infty$, and $f(z)$ is meromorphic on $\overline{\mathbb{C}}$ hence rational. This seems a bit too sketchy. I think that $\frac{1}{f(z)-w_0}$ is bounded at $\infty$ by taking $\lim_{z \rightarrow \infty} f(z)$. How is $f(z)$ meromorphic on $\overline{\mathbb{C}}$? Also, how does $f(z)$ attain values $w$ near $w_0$ only near points in $f^{-1}(w_0)$. I understand that it is the maximum. However, I don't see precisely why. Thanks.