Prove that the limit of zeroes of a non zero complex valued function is an essential singularity of the function

Question

Theorem : If f is a non zero complex valued function, the limit point of zeroes of f is an essential singularity of f.

I need a proof of this theorem.

Answer 1

If $z_0$ is a pole of a $f$ then necessarily $|f(z)| \to \infty$ as $z \to z_0$. [Rudin's book has a proof of this fact]. Hence a limit point of zeros cannot be a pole. It cannot be a removable singularity also so it must be an essential singularity.