Prove that the $k^{th}$ derivative of $f$ has necessarily infinitely many zeros

Question

I have the following question: Let $f$ be a real entire function, i.e., $$f(x)=∑_{n=1}^{∞}a_{n}x^{n}$$

with infinitely many zeros.

Prove that the $k^{th}$ derivative of $f$ has necessarily infinitely many zeros for all $k≥1$.

Answer 1

hint use the mean value theorem between adjacent zeroes, and induction