# The image of a meromorphic function is dense

Claim 1: The image of the complex plane under a nonconstant entire function is dense in the complex plane.

Claim 2: The image of any small enough punctured neighborhood around an essential singularity is dense in the complex plane.

Claim 3: The image of the complex plane under a nonconstant meromorphic function is dense in the complex plane.

I know the proof to the first claim and it uses Liouville's theorem. I know the proof to the second claim (the Casorati-Weierstrass theorem)

I'm trying to prove claim 3 but I don't have any direction. I have tried proving it by way of contradiction, just like how the first 2 claims are proved but got stuck.. Please don't give me a full solution, hints will suffice :)

• If $f$ avoids the open ball $B_r(z_0)$, then $z\mapsto \frac1{f(z)-z_0}$ is bounded by $\frac1r$ - and what happens to the poles? – Hagen von Eitzen May 22 '17 at 11:29
• The poles become zeros of the new function. So can we say the new function is entire, bounded and therefore constant? – zokomoko May 22 '17 at 13:50