Convergence uniformly in compacts and zeros in D(0,1)

Question

I have a problem figuring this out.

If I have a secuence of holomorphic functions $\{f_n\}_{n\ \in\ \mathbb{N}}$, and none of them have a zero in $D(0,1)$. That secuence converges uniformly in compacts to the function $f$ in the disc $D(0,1)$. I am asked if this statements are true or false:

1. $f$ has no zeros in $D(0,1)$.
2. $f(z)=0,\ \forall z\in D(0,1)$
3. $f$ has exactly one zero in $D(0,1)$.

My guess: Because the functions in the secuence $f_n$ are holomorphic, the function they converge to ($f$) has to be holomorphic, and also $f_n^{(k)}\rightarrow f^{(k)}$ unif. in compacts.

But I'm stuck there. I think 3. is true, but I do not know how to continue. Any guesses/hints?

Answer 1

Hurwitz's Theorem says that if $f_n$ is a sequence of holomorphic functions on a domain $D$ which converges normally (i.e.uniform convergence on each compact subset of $D$) to $f$.If each $f_n$ does not have a zero on $D$ then either $f$ is identically zero or also is nowhere zero.