Suppose that we have a sequence of holomorphic non-constant functions $(f_{n}(z))_{n=1}^{\infty}$ and $F(z)$. These functions are defined on a connected open set $A\subset \mathbb{C}$ that includes imaginary axis.
I see that $f_{n}$ converges uniformly to $F$ on any compact subset of $A$
Additionally $$\lim_{n\to \infty}\sup\{|Re(s)|:f_{n}(s)=0, s\in A\}=0$$
Where $Re(s)$ is a real part of complex number $s$
My question:
Do the statements above imply that all the zeros of $F(z)$ are purely imaginary. This is $F(z)=0\implies Re(z)=0$ ? If not, what is conterexample and how the assumptions should look like?
I have an intuition that zeros of these functions move closer to the imaginary axis just like the whole functions, but i don't know how to prove it.
Any help will be appreciated.