Possible error at definition of essential singularity Let $\Omega \subseteq \mathbb{C}$ be open and $a\in\Omega$, with $f\in\mathcal{H}(\Omega\backslash\{  a\})$.
A singularity at $a$ is called essential, iff $\forall\, w\in\mathbb{C},\ \exists$ a sequence $(z_n)_{n}$ in $\mathbb{C}$ with $\lim_{n\to\infty}z_n=a$, so that $\lim_{n\to\infty}f(z_n)=w$.
I know that there are quite a few different defintions of an essential singularity, but they all amount to the fact, that with the possible exception of a point, every complex value is adopted in an arbitrary small neighborhood around the singularity. (Great Picard theorem) At least, if I am correct.
Now how does this first definition in my post make sense? This just seems completely wrong to me. I might live with something like 
"A singularity is called essential, iff $\forall\, w\in\mathbb{C},\ \exists$ a sequence $(z_n)_{n}$ in $\mathbb{C}$ with $\lim_{n\to\infty}z_n=c,\, \ c\in\mathbb{C}$, so that $\lim_{n\to\infty}f(z_n)=w$."
But if $a$ is fixed and the singularity, how can this be true?

If $f\in\mathcal{H}(\Omega\backslash\{  a\})$, then $a$ is called a isolated sinfgularity of $f$.
The singularity is $a$ is called essential, if $a$ is not a pole and $a$ is not a removable singularity of $f$.
THEOREM(Casorati- Weierstraß): The singularity $a$ is essential iff $\forall\, w\in\mathbb{C},\ \exists$ a sequence $(z_n)$ in $ \Omega \setminus \{a\}$ with $z_n \to a$ and $f(z_n) \to w$.
