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Say we have a region $D$ and a sequence of functions $f_{n}$ holomorphic in $D$, which converges uniformly on compact sets to a one-to-one function $f$. Can we say that for each compact set $K \subset D$ there is a number $N(K)$ such that $f_{n}$ is one-to-one for all $n >N(K)$?

Thank you for any help or suggestions.

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How about using Hurwitz theorem, this gives you a ball around each point in $K$ in which $f_n$ and $f$ have the same zeroes, now use compactness of $K$.

Edit: False if global injectivity of the $f_n$ is required: $f=z$, $f_n=z+(z^2/n)$.

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    pel: Yes, I believe so too, and, just for the record, my argument doesn't break down because one of the zeroes of $f_n$ has to tend to infinity as $n\rightarrow \infty$, so maybe at least a partial converse is true (in compact subsets with simple zeroes).2011-04-15