Let $\displaystyle \{ f_n(z) \}$ be a sequence of rational functions that converges normally to $\displaystyle f(z)$ on the extended complex plane $\displaystyle \mathbb{C}^*$. Show that $\displaystyle f_n(z)$ has the same degree as $\displaystyle f(z)$ for $\displaystyle n$ large.

I believe that the degree of a rational function is the maximum of the degrees of its constituent polynomials.

The back of the book says to show $\displaystyle f_n(z)$ eventually has same number of zeros as $\displaystyle f(z)$. I do not see use this to prove the problem. I need some help on this problem. Thank you.