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Let $\Omega$ be a zone, $f_n \in H(\Omega)$, $n=1,2,3,\ldots$ and functions $f_{n}$ don't have zero in $\Omega$ and the chain $(f_n)$ converges uniformly towards $f$ over the complex subset of $\Omega$. Verify that $f$ either doesn't have zeroes inside $\Omega$ or $f(z)=0$, for all $z \in \Omega$.

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closed as off-topic by Jonas Meyer, Did, N. F. Taussig, graydad, hardmath Apr 1 at 16:10

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Welcome to MSE! Please typeset the question clearly. Use $...$ only for wrapping formulas, not complete sentences. –  lhf Jul 3 '12 at 12:29
Do you mean uniform convergence over the compact subsets of $\Omega$? –  Davide Giraudo Jul 3 '12 at 12:31
zone is probably region, chain is probably sequence. –  lhf Jul 3 '12 at 12:33
What have you tried? If this is homework, tag it as such. –  lhf Jul 3 '12 at 12:58
... and complex is probably compact. –  GEdgar Jul 3 '12 at 14:20

1 Answer 1

This is called Hurwitz's Theorem. Wikipedia has a (not particularly well-written) webpage on this at http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(complex_analysis) It's also in standard textbooks like Ahlfors and so on.

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