Complex Analysis proof question (from Markushevich text))

1. Jan 2, 2012

imurme8

There is a proof offered in the text "Theory of Functions of a Complex Variable" by Markushevich that I have a question about. Some of the definitions are a bit esoteric since it is an older book. Here "domain" is an open connected set (in $\mathbb{C}$, in this case.)
The proof that $f(\gamma)\subseteq \Gamma$ is clear to me. For the opposite containment, they offer the following proof:
My question is about the part marked with an asterisk. How do we know that $z_{n}$ has a convergent subsequence? If $\{z_n\}$ were bounded, then I see that we would have this result by the Bolzano-Weierstrass theorem. Also if we were working in the extended complex plane, then compactness of $\tilde{\mathbb{C}}$ would give us this result. But here, I don't see any indication that either $G$ or $f(G)$ is bounded, so I don't see how we have this.

Just reading this again, it strikes me that perhaps we can bound $\{z_n\}$ by the continuity of the inverse function $f^{-1}$. But I don't see that this continuity is assumed, so I'm not sure.

Thanks for any help!

2. Jan 7, 2012

imurme8

On second thought, I think the proof works as long as we suppose that we're working in the extended complex plane.