This is an exercise problem from Complex Analysis by Joseph Bak and Donald J. Newman.
Suppose $f$ has an isolated singularity at $z_0$. Show that $z_0$ is an essential singularity if and only if there exist sequences $\{\alpha_n\}$ and $\{\beta_n\}$ with $$ \alpha_n \to z_0, \quad \beta_n \to z_0,\qquad\text{ and }\qquad f (\alpha_n) \to 0, \quad f (\beta_n) \to \infty.$$
I know Casorati–Weierstrass' theorem or Picard's theorem should guarantee that if $f$ has an essential singularity, then such suitable sequences should exist such that $f$ can approach any value, which we can set to be $0$ or $\infty$.
I, however, am not sure how to prove the converse. I'm wondering if anyone can offer a hint. Thanks.