HurwitzsTheorem

Hurwitz's Theorem

Let ${f_n}$ be a sequence of holomorphic functions converging uniformly to a function $f$ on a compact subset $D$ of the complex plane, and let $z_\infty$ be a zero of $f$ of order $m$. If each $f_n$ has a zero of order $m$ in $D$ for all sufficiently large $n$, then there exists a sequence of points ${z_n}$ in $D$ with the following properties:

1. For each $n$, $z_n$ is a zero of $f_n$ of order $m$.

2. The sequence ${z_n}$ converges to $z_\infty$ as $n$ approaches $\infty$.