I'm stuck on the following problem:
Let $f_n$ be a sequence of injective analytic functions on the unit disc $D$ such that $f_n$ converges uniformly to $f$ on compact subsets of $D$. Show that $f$ is either injective or constant.
Already $f$ is analytic as an almost uniform limit. If $f'$ is identically zero, then $f$ is constant. Otherwise $f'$ has isolated zeroes. If $c$ is such an isolated zero, then $f_n'$ converges uniformly to $f'$ on a small disc near $c$. By Hurwitz's theorem, for almost all $n$ the functions $f_n'$ should also have a zero in that small disc.
But an analytic function is locally injective at a point if and only if its derivative at the point is nonzero. And injective implies locally injective. So the $f_n'$ shouldn't have any zeros in $D$.
So, we can at least conclude that $f'$ is never zero on $D$. This shows that $f$ is locally injective. Is there an easy way to conclude injectivity from local injectivity here?