What's a good reference for the continuity of the number of zeros?

Question

What is a good reference for the following statement, or something that easily implies it?

For all sequences $\:\langle\:f_0,f_1,f_2,f_3,...\rangle\:$ of (complex) analytic functions $\:\:f_n : \mathbb{C} \to \mathbb{C}\:$,
for all complex analytic functions $\:\:f : \mathbb{C} \to \mathbb{C}\:$, $\:$ if $\:\:f(0) = 0\:$ and $\:f$ is not identically zero
and $\:\langle\:f_0,f_1,f_2,f_3,...\rangle\:$ converges uniformly to $\:f$ on the unit disk,
then for sufficiently large $n$, $\:f_n$ has a zero in the unit disk.


I saw this proved back in complex analysis and could probably re-prove it from
Cauchy's integral formula and the argument principle, but I feel that there should
be some reference that can be used, even though I haven't been able to find one.

Answer 1

This is called Hurwitz's theorem; the Wikipedia page has references.