I am having a lot of difficulty with the following past qualifying exam problem. Any help would be awesome. Thanks.
Let $\Omega_1 = \mathbb{C}\setminus \left \{\{0\} \cup \{\dfrac{1}{n}:n\in \Bbb N\}\right \}$ and $\Omega_2 = \mathbb{C}\setminus\{z : \text{Im } z=0,\ |\text{Re} z|\ge 1\}.$ Construct a non-constant analytic function $f : \Omega_1 \to \Omega_2$ or show that this is impossible.
$\Omega_2$ is biholomorphic to the open unit disc, so the problem is equivalent to constructing a non-constant holomorphic function $g$ on $\Omega_1$ with absolute value $<1$. Now each of $1/n$ would be an isolated singularity of $g$, and as $g$ is bounded, each of them is removable, i.e. $g$ extends to $\mathbb C\setminus\{0\}$. But then $0$ is an isolated singularity for this extended function which is still bounded, so $g$ extends to an entire bounded function, hence $g$ is constant by Liouville theorem.