Uniform limit of injective analytic functions is injective

Uniform limit of injective analytic functions is injective, if this limit IS NOT CONSTANT.

Proof. Suppose that $f(z)=a$ at two points $z_1,z_2$. Surround these two points by a Jordan curve $\gamma$ such that $f(z)\neq a$ on $\gamma$. Then according to the argument principle $$\frac{1}{2\pi i}\int\frac{f'(z)dz}{f(z)-a}\geq 2.$$ (The right hand side equals to the number of solutions of $f(z)=a$ inside $\gamma$, counting multiplicity). On the other hand, $f_n(z)\neq a$ on $\gamma$ when $n$ is sufficiently large, and the similar integral with $f_n$ instead of $f$ is $\leq 1$, because $f_n$ are injective. But this is a contradiction because $f_n\to f$ uniformly, so integral with $f_n$ must converge to the integral with $f$.