Casorati-Weiestrass theorem for essential singularities
Let $\Omega\subseteq \mathbb{C}$ be open, $ a\in\Omega,\ f\in H(\Omega\backslash \{a\})$ ($f$ analytic on $\Omega\backslash \{a\})$. In the case of an essential singularity: If $ C(a,r)\subseteq\Omega$ ($C(a,r)$ is the disk with origin $a$ and radius $r$), then $f(C(a,r)\backslash \{a\})$ is dense in $\mathbb{C}$.
$\color{green}{\text{(1) What does this theorem actually say? How would you put in descriptive words?}}$
$\color{green}{\text{(2) How would an outline of the proof (in words) look like? What steps should be followed?}}$