I am just looking at normal families of functions and they have made me somewhat complex to understand, in particular I have two doubts that I consider essential to both distinguish and understand how they are and these are the following:
A) Is there a normal family of holomorphic functions $F$ that has a sequence without subsucessions that converge uniformly in a domain? That is, can you give an example of a normal family of holomorphic functions in the domain $ U $ and a sequence $ \{f_n\} $ contained in $ F $ that does not have a subsucession that converges uniformly in $ U $? Or must they all necessarily have convergent subsucessions in U?
B) Can an example be given of a normal family of holomorphic functions in $U$ domain and a sequence $\{f_n\}$ contained in $F$ that converges uniformly in compacts of $U$ to a point $f$ and such that $f$ is not in $U$?
I hope you can help me please, thank you very much in advance.
For part B), I think this answer works, but I wonder if I am correct.
(A): $U=\{z: |z|<1\}, f_n(z)=z^{n}$. This sequence is normal but no subsequence converges uniformly on $U$.
[This sequence converges uniformly on compact subsets of $U$ to $0$ but not on $U$: Suppose $|z^{n_k}| <\frac 1 2$ for all $z \in U$, for all $k$ sufficiently large. Let $z \to 1$ to get a contradiction].
(B) : The link you have given works.