if a series of polynomial converge uniformally locally, and each of then has only real roots, then if f is an entire, then has only real roots.

Question

let $P_N$ be a series of polynomial that has only real roots, which converges uniformly locally. let the function f be the function they converge into, And let's assume that f is an entire function. A) prove that f has only real roots (if it is not the 0 function). B) let $f^{(k)}$ be not the zero function. Prove then that $f^{(k)}$ has only real roots.