I'm trying to understand the proof of theorem 24.1 from the book Analytical theory of continued fractions by H.S. Wall. The theorem states that
Let $\{f_p(z)\}$ be a sequence of functions, analytic in a simply connected open domain S which is uniformly bounded over every closed domain entirely within S. Then there exists an infinite subsequence of these functions which is uniformly convergent over every finite closed domain entirely within S to a limit function which is analytic in S.
I understood everything up until the last paragraph of the proof. Here is an excerpt from the book:
I think this proof makes a tautological statement. Namely they assume that the quantity $$ n(\epsilon/5,\zeta)\tag{1} $$ is finite.
Question: Is this true $$ N=n(\epsilon,\zeta)? $$
If Yes then what they proved is a tautological statement $N=n(\epsilon,\zeta)\leq n(\epsilon/5,\zeta)\leq \infty$ This does not mean that $n(\epsilon,\zeta)$ can not be infinite.
Am I missing something here?
If you don't know answer to this question maybe you can suggest an alternative source for the proof of the theorem above?
It is fine to assume that $n(\epsilon/5, \zeta)$ is finite, because it cannot be infinite.
We $n(\epsilon, z)$ to be the least positive integer such that (24.3) holds. I am going to assume that at some point before the excerpt you quoted, it is shown that the sequence $F_p(z)$ converges to some limit $F(z)$ pointwise, and therefore (24.3) must hold at some integer. The set of all integers where (24.3) holds has a (finite) least element.
Meanwhile, $N$ is not necessarily finite, because $N$ is defined as a limit: $$N = \lim_{p \to \infty} n(\epsilon, z_p).$$This limit could be infinite, even if every $n(\epsilon, z_p)$ is finite. It is only once we show that the sequence $n(\epsilon, z_p)$ has an upper bound that we can conclude that $N$ must be finite as well.