Hurwitz's theorem

Statement

Let $G$ be a domain in $\mathbb{C}$, that is, a connected open set. Let $(f_k)$ be a sequence of holomorphic functions on $G$ that converges uniformly on each compact subset of $G$ to a function $f$. Suppose that $f$ is not identically zero on $G$, and that $f$ has a zero of order $m$ at a point $z_0 \in G$. Then for each sufficiently small $\rho > 0$ there is an integer $N$ with the following property. For each $k \ge N$, the function $f_k$ has exactly $m$ zeros in the disc $|z - z_0| < \rho$, each counted as many times as its order.

Remarks

The limit $f$ is holomorphic on $G$ by the theorem on uniform limits. A consequence: if no $f_k$ has a zero in $G$, then $f$ either has no zero in $G$ or is identically zero. A second consequence: if each $f_k$ is injective on $G$, then $f$ is injective or constant. The radius $\rho$ must be small: $f_n(z) = z - 1 + \frac{1}{n}$ has one zero in the unit disc, but its limit $z - 1$ has none there.

Explanations

L1 human

Zeros of holomorphic functions do not suddenly appear or disappear under uniform limits. Near a zero of the limit, each function far along in the sequence has the same number of zeros. The proof compares $f_k$ with $f$ on a small circle, as in Rouché's theorem.