complex numbers- hurwitz theorem

Suppose that D is a connected set open set , $\displaystyle f_{n} \in H(D)$ and

$\displaystyle f_{n} -> f $ uniformly on compact subsets of D. If f is nonconstant and

$\displaystyle z \in D $ then there exists N and a sequence $\displaystyle z_{n} -> z$ such that $\displaystyle f_{n} ( z_{n} ) = f(z)$ for all $\displaystyle n \geq N$

Hint: Assume that f(z) = 0. Apply the hurwitz theorem in the disk $\displaystyle D(z, r_{j}) $ for a suitable sequence $\displaystyle r_{j}->0 $

I didn't even understand the hint.hurwitz theorem states that $\displaystyle f_{n}$

and f have the same number of the zeroes in the disk. Can anyone help, thanx.