Couple of questions on Hurwitz theorem

Question

Hurwitz theorem as stated in Hahn and Epstein's Classical Complex Analysis is as follows:

Answer 1

For (2), assuming you mean real-analytic $f\colon\mathbb{R}\to\mathbb{R}$: take $f_n(x)=x^2+\frac1n$, which has no real roots, converging to $f(x)=x^2$ which has a (double) root $x=0$.

For (1), yes. Let $f_n(z)=f(z)+a_n$, $a_n\to 0$ and Rouché's theorem to get a root of $f_n$ sufficient close $z_0$.

Answer 2

Here's a related question.

Let $\{f_n\}$ be a sequence of analytic functions on a domain $D$ converging uniformly on every compact subset of $D$ to a function $f$. Suppose that $f$ has a simple zero $z_0 \in D$. Show that there is a neighborhood of $z_0$, say $N(z_0) \subset D$ and an integer $n_0>0$ such that for each $n > n_0$, $f_n$ has a simple zero in $N(z_0)$.