Hurwitz's Root Theorem

Let $\{f(x)\}$ be a Sequence of Analytic Functions Regular in a region $G$, and let this sequence be Uniformly Convergent in every Closed Subset of $G$. If the Analytic Function

$$\lim_{n\to\infty} f_n(x)=f(x)$$

does not vanish identically, then if $x=a$ is a zero of $f(x)$ of order $k$, a Neighborhood $\vert x-a\vert<\delta$ of $x=a$ and a number $N$ exist such that if $n>N$, $f_n(x)$ has exactly $k$ zeros in $\vert x-a\vert<\delta$.

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 22, 1975.