Define the ball at $z_{0}$ of radius $r$ as $B(z_{0},r)=\{z\in G:\lvert z-z_{0}\rvert<r\}$ and $D(z_{0},r)=\{z\in G:\lvert z-z_{0}\rvert\leq r\}$ is the closed ball at $z_{0}$ of radius $r$.

What this theorem says is that if you have a sequence of holomorphic functions which converge uniformly on compact subsets (such a sequence always converges to a holomorphic function but that’s another theorem altogether), the limit function is not identically zero and furthermore the limit function is not zero on the boundary of some ball, then eventually the functions of the sequence have the same number of zeros inside this ball as does the limit function.

Do note the requirement for $f$ not being identically zero. For example the sequence $f_{n}(z):=\frac{1}{n}$ converges uniformly on compact subsets to $f(z):=0$, but $f_{n}$ have no zeros anywhere, while $f$ is identically zero.

Also in general this result holds for bounded convex subsets but it is most useful to state for balls.

An immediate consequence of this theorem is this useful corollary.