Problem
Let $f$ be a non-constant entire function. Suppose that there is a sequence of polynomials ${P_n(z)}$, $n=1,2,...$ such that
- $P_n(z)$ converges uniformly to $f$ on every bounded set in $\mathbb{C}$
- All zeros of $P_n$ are real.
Prove that all zeros of $f$ are real
Progress
I have tried Hurwitz's theorem but I could not figure out how to go after that. By Hurwitz, I know that the sequence and $f $ have the same number of zeros in $B(a;R)$. But that's about it. Also, I think the sequence of zeros of $P_n$ converge to the zeros of $f$, with respect to their magnitude and multiplicity. But I really do not know how to show it.
Suppose $u$ is a non real zero of $f$. There exists $R>0$ such that the closed disk with center $u$ and radius $R$ do not intersect the real line, and such that $f(z)\not =0$ on the circle $C$ centered at $u$ and of radius $R$. Then there exists $m>0$ such that $|f(z)|\geq m$ on $C$. Now there exists a large integer $N$ such that $\displaystyle |f(z)-P_N(z)|<\frac{m}{2}<m\leq |f(z)|$ on the circle $C$. By Rouché's theorem, $f$ and $P_N$ have the same number of zeroes in the disk $D(u,R)$. As $f$ have at least $u$ as zero, $P_N$ has a zero in $D(u,R)$, hence non real, contradiction.