Suppose $\Omega\subset\mathbb{C}$ is a region, $f_n\in H(\Omega)$, for $n=1,2,3,...,$ none of the functions ${f_n}$ has a zero in $\Omega$, and $\{f_n\}$ converges to $f$ uniformly on compact subsets of $\Omega$. Prove that:
(1)either f has no zero in $\Omega$ or $f(z)=0$ for all $z\in\Omega$
(2)if \Omega' is a region that contains every $f_n(\Omega)$, and if $f$ is not constant, then f(\Omega)\subset\Omega'
Luckily, I can prove the first question, using Rouche's Theorem. However, I really have no idea how to prove the second one, because I don't know what "$f_n(\Omega)\subset\Omega'$" implies. It seems that I have few tools to deal with that, based on theorems I learnt from my textbook.
PS: I cannot see why \Omega' should be a region.
Any hints will be very appreciated. Actually, I just want some hints. I believe that this question is very easy.