Complex analysis: Prove a meromorphic function to be rational. I come across a problem about complex analysis:
Show that a meromorphic function on the complex plane, which achieves any complex number no more than fixed given times, must be rational.
The only way I know to prove a meromorphic function being rational is to show the infinity is not an essential singularity of the function (thus it can be controlled by polynomial). Following this, we can use Picard's Great Theorem to solve the problem.
I wonder if anyone can help me think of another method. (Without using Picard's Great Theorem.)

Take the extended complex plane $\mathbb{C}\cup\{\infty \}$. Introduce a function $f$ that has a punctured neighbourhood at the origin (hence is analytic in a punctured neighbourhood at infinity). The remaining region has finitely many poles hence the collection of poles of $f$ (denoted $a_n$) is finite. Introduce Laurent expansions at one of the poles $a_m$ (for fixed $m$) and at infinity. Look at the nagtive power sections of Laurent expansions at the pole and infinity (call these $l_k$ and $l_{\infty}$ respectively) to derive polynomials of finite length at each pole, and subtract them from $f$. This will give something like
\begin{equation*}
f-\sum l_k-l_{\infty}.
\end{equation*}
Conclude that this result is entire and bounded to get that this is rational.
