I need someone to decide whether I'm going crazy. Dettman states the following theorem without proof:
Theorem 4.2.5 Let $F$ be a family of functions analytic in a domain $D$, where it is uniformly bounded. Then every sequence of functions in $F$ contains a subsequence which converges to a function in the family.
Now, as stated, I think the theorem is incorrect. As a counter-example, take $F = (z^n)_{n\in \mathbb{N}}$ on the open unit disk. As a sequence, these functions converge to 0, which isn't in the family.
So here's my question: Is this theorem unsalvageable, or can it be perturbed to a true statement about normal families? The theorem of course becomes true if we omit 'in the family,' or if we replace it with 'which is also analytic and bounded in $D$.' But is there a similar (true!) theorem which guarantees membership in the family?
Indeed, as stated, the theorem is incorrect.
There are two ways to change it to obtain a true theorem, one can weaken the conclusion - here, one drops the erroneous assertion that the limit necessarily belongs to the family - or one can strengthen the premises. Since the space of holomorphic functions is metrisable (in particular, it is a sequential space) in the topology of locally uniform convergence, the limits of all convergent sequences in a subset belong to the subset itself if and only if it is closed, so we can strengthen the premises to require a closed (in the topology of locally uniform convergence) family $F$.