A holomorphic function $f$ has an essential singularity in $0$ iff $\exists(z_k)_k$ s.t. $z_k\to 0$ and $|z_k^mf(z_k)|\to\infty$ for all $m$

Question

Let $f:\mathbb{C}\setminus\left\{0\right\}\to\mathbb{C}$ be a holomorphic function $\Rightarrow$ $f$ has an essential singularity in $0$ if and only if $\forall m\in\mathbb{N}:\exists (z_k)_{k\in\mathbb{N}}\subset\mathbb{C}\setminus\left\{0\right\}$ such that $z_k\to 0$ and $\left|z_k^mf(z_k)\right|\to\infty$ for $k\to\infty$.

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Since $f$ is holomorphic $$f\equiv\sum_{k=-\infty}^\infty a_kz^k$$ for some $a_k\in\mathbb{C}$. Moreover, $f$ has an essential singularity in $0$ if and only if $\exists\left(k_j\right)_{j\in\mathbb{N}}$ such that $k_j\to -\infty$ and $a_{k_j}\ne 0$.

How does one prove the theorem given these facts?

Answer 1

For one direction, if $f$ has a removable singularity or a pole of order $k$ in $0$, what can you say about the function $z\mapsto z^m f(z)$ for large enough $m$?

For the other direction, note that if $f$ has an essential singularity in $0$, then so has $z \mapsto z^m f(z)$ for all $m \in \mathbb{Z}$. Appeal to the Casorati-Weierstraß theorem if necessary.