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Complex Analysis proof question (from Markushevich text))

  1. Jan 2, 2012 #1
    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 [itex]\mathbb{C}[/itex], in this case.)
    The proof that [itex]f(\gamma)\subseteq \Gamma[/itex] 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 [itex]z_{n}[/itex] has a convergent subsequence? If [itex]\{z_n\}[/itex] 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 [itex]\tilde{\mathbb{C}}[/itex] would give us this result. But here, I don't see any indication that either [itex]G[/itex] or [itex]f(G)[/itex] is bounded, so I don't see how we have this.

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

    Thanks for any help!
  2. jcsd
  3. Jan 7, 2012 #2
    On second thought, I think the proof works as long as we suppose that we're working in the extended complex plane.
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