Let $f$ be a holomorphic, non-constant function on $D=\{0<|z|<10\}$. It is given that for every $n \in \mathbb{N}$, $|f(\frac{1}{n})| \le \frac{1}{n!}$.
Show that $f$ has an essential singularity at 0.
I have no idea how to even approach this. Why is it not possible for $f$ to have a pole at 0?
Note. In the original question it was written $n \in \mathbb{C}$ but I considered this a typo.
Thanks in advance!