Prove or disprove A nonconstant analytic function can have a

Prove or disprove: A non-constant analytic function can have at most finitely many zeros on a closed disk.

Solution

Suppose an analytic function f has infinitely many zeros on some closed disk D. Then there exists a sequence of zeros in D with a limit point in D. Thus by the identity theorem (Let D be a domain and f analytic in D. If the set of zeros Z(f) has a limit point in D, then f 0 in D.), f is identically zero and thus constant.

so for a non-constant analytic function must have finite zeros on a closed disk