Let $f$ be holomorphic in $\mathbb{C} \setminus \{0\}$.

Prove or disprove: If $|f(\frac{i}{n})| \ge n$ for all $n \in \mathbb{N}$, then $f$ has a pole in $0$.

The equation yields that $f$ is not bounded near $0$, from which I know it is not a removable singularity. So it's either a pole or an essential singularity.

I have tried to use [Casorati-Weierstraß theroem](https://en.wikipedia.org/wiki/Casorati%E2%80%93Weierstrass_theorem), but I couldn't make any sense of it.