4. Theorem 2.10.2 - Theorem of Hurewicz

Let $G \subseteq C$ be a domain, and $f_{n} : G \to C$ be a sequence of functions converging locally uniformly to a function $f : G \to C$ . If each $f_{n}$ has less than $m$ zeros with multiplicity, then either $f$ 1;;has less than $m$ zeros with multiplicities, or $f$ is zero in $G$ .

PROOF:

proof by contradiction
Find a compact set containing all the points
use existence of minimum.
at some point apply theorem of Rouche

Remark

I didn't understand how we applied the theorem of Rouche at the end.