Let $G$ be a region in $\mathbb{C}$ and $a\in G$. Suppose $f:G-\{a\}\to\mathbb{C}$ is an injective analytic function such that $f(G-\{a\})=\Omega$ is bounded. Show that $f(a)\in\partial\Omega$.
I know a couple things. Since $f$ is injective it's non-constant, so by the open mapping theorem we know $\Omega$ is open. Also, $f$ is obviously bounded in a neighborhood of $a$, so $z=a$ is a removable singularity. Then we can define $f$ so that it's analytic at $a$, possibly losing injectivity in the process. But I'm stuck here.
Any help is appreciated