Complex Analysis proof question (from Markushevich text))

[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.)

Theorem: Let G be a domain with boundary \gamma, let f(z) be a one-to-one function on G, and let f(z) be continuous on \overline{G}. Then f(\gamma)=\Gamma, where \Gamma is the boundary of the domain f(G).

The proof that f(\gamma)\subseteq \Gamma is clear to me. For the opposite containment, they offer the following proof:

Suppose w \in \Gamma. Then there exists a sequence \{ w_n \} such that w_n \in f(G) and w_n \to w as n \to \infty. If z_n is the inverse image of w_n, then z_n \in G, and from the sequence \{z_n\} we can select a subsequence \{z_{n_k}\} (*) converging to a point z\in \overline{G}, where

f(z)= \lim\limits_{n\to \infty} f(z_{n_k}) = \lim\limits_{n\to \infty}w_{n_k}=w.​

Thus w is the image of a point z\in \overline{G}. But z \notin G, since, according to a theorem, the image of an interior point of G is an interior point of f(G) (since f is one-to-one). Therefore z is a boundary point of G and w=f(z) belongs to f(\gamma). It follows that f(\gamma)\supseteq \Gamma, and the proof is complete.

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!

 

[imurme8]

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