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Let $\Omega$ be open and connected. Suppos that a sequence $$\{f_n : \Omega \to \mathbb{C} : f_n \mbox{ is holomorphic and injective in $\Omega$ }\}$$ Converges uniformly to a function $f$ on every compact subset of $\Omega$. Prove that if $f$ is not a constant in $\Omega$, then $f$ is injective in $\Omega$.

First I proved that $f$ is holomorphic.

Suppose that $f$ is not injective.

There are distinct point $z_1,z_2$ in $\Omega$.

Then there are path from $z1$ to $z2$ lying in $\Omega$. Then... I can't tell you How to solve.

Could you help me?

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marked as duplicate by Martin R, mrf complex-analysis Aug 2 '16 at 9:37

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  • $\begingroup$ Do you know Hurwitz theorem ? $\endgroup$ – C. Dubussy Aug 2 '16 at 9:07
  • $\begingroup$ Are you allowed to use the open mapping theorem? It would help here. $\endgroup$ – Joey Zou Aug 2 '16 at 9:09