$f$ is analytic in the unit circle, except a simple pole. Prove that $\displaystyle \lim_{n\to\infty} |a_{n}|>0$

Question

I'm learning complex analysis and encountered this question:

Denote $\mathbb{D}$ to be the unit circle. $f$ is analytic in ${\mathbb{D}}\subseteq U$ except a simple pole $z_{0}\in \partial\mathbb{D}$. Let $\displaystyle f=\sum_{n=0}^\infty a_{n}z^n$ be the Taylor representation of $f$ around $z=0$.

Prove that $\displaystyle \lim_{n\to\infty} |a_{n}|>0$ and compute $\displaystyle \lim_{n\to\infty}\frac{a_{n+1}}{a_{n}}$.

My attempt: Write $\displaystyle f(z)=\frac{g(z)}{z_{0}-z}$ and using Taylor representations of both understand $a_{n}$ better. I got stuck though, so I would love some help.

And follow-up questions: Can this be generalized to poles (not just simple ones)?