Say we have a sequence of entire functions $g_{n}$, and $g_{n} \rightarrow g$ uniformly on compact sets in $\mathbb{C}$. I want to show that if each $g_{n}$ has only real zeros, then all the zeros of $g$ must also be real.
My best guess was to assume $g$ has a non real zero $w$ and take a compact set $K$ containing no real numbers and containing $w$. Then $g_{n}$ never vanishes on $K$ yet $g$ does. What if we look at the the sequence of functions $\frac{1}{g_{n}}$ analytic on $K$. Is it true $\frac{1}{g_{n}} \rightarrow \frac{1}{g}$ uniformly on compact sets? Because then we would get $\frac{1}{g}$ is analytic in $K$, a contradiction. Thanks for the help!