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Let G be a simply connected domain, $G \not \neq \mathbb{C}$ and $z_0 \in G$, I got to show that for every $n \in \mathbb{N}$ there is a holomorphic and injective mapping: $f_n:G \rightarrow D_1(0)$, the open unit disk. That is $f_n(z_0)=1-\frac{1}{n}$. Then I got to show that this $f_n$ converges to locally uniformly to a non-injective function.

I think the first assertion I can show with the Riemann mapping theorem, right? I just I don't understand why it connverges to a non-injective function, as all $f_n$ are biholomorphic by the Riemann mapping theorem. Am I totally wrong?

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  • $\begingroup$ If $f_n\to f$ what is $f(z_0)$? Now what does the Maximum Modulus Theorem say about $f$? Or the Open Mapping Theorem? $\endgroup$ – David C. Ullrich Jun 23 '16 at 18:40
  • $\begingroup$ that this function must be constant. so it would be f=1? $\endgroup$ – Thesinus Jun 23 '16 at 18:48

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