Let’s first state it.
Theorem (Hurwitz’s theorem). Suppose is a sequence of analytic functions on a domain that converges normally on to , and suppose that has a zero of order at . Then for every small enough , there is large such that has exactly zeros in the disk , counting multiplicity, and these zeros converge to as .
As a refresher, normal convergence on is convergence uniformly on every closed disk contained by it. We know that the argument principle comes in handy for counting zeros within a domain. That means
The number of zeros in , arbitrarily small, goes to the number of zeros inside the same circle of , provided that
To show that boils down to a few technicalities. First of all, let be sufficiently small that the closed disk is contained in , with inside it everywhere except for . Since converges to uniformly inside that closed disk, is not zero on its boundary, the domain integrated over, for sufficiently large . Further, since uniformly, so does , so we have condition such that convergence is preserved on application of integral to the elements of the sequence and to its convergent value. With arbitrarily small, the zeros of must accumulate at .