Let \{ f_n(z) \} be a sequence of rational functions that converges normally to f(z) on the extended complex plane \mathbb{C}^*. Show that f_n(z) has the same degree as f(z) for 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 f_n(z) eventually has same number of zeros as f(z). I do not see use this to prove the problem. I need some help on this problem. Thank you.