In Ahlfors' text of Complex Analysis, chapter 5 theorem 1, he proves the following:
Theorem I. Suppose that $f_n(z)$ is analytic in the region $\Omega_n$, and that the sequence $\{f_n(z)\}$ converges to a limit function $f(z)$ in a region $\Omega$, uniformly on every compact subset of $\Omega$. Then $f(z)$ is analytic in $\Omega$. Moreover, $f_n'(z)$ converges uniformly to $f' (z)$ on every compact subset of $\Omega$.
In the proof he examines a closed disk $|z-a| \leq r$ which is contained in $\Omega$. He claims that from the assumptions of the theorem the disk lies in all $\Omega_n$ for $n>n_0$ for some $n_0$.( I think this is clear, due to the uniform convergence on compact subsets.) Under this sentence there is a footnote which states:
In fact, the regions $\Omega_n$. form an open covering of $|z - a| \leq r$. The disk is compact and hence has a finite subcovering. This means that it is contained in a fixed $\Omega_{n_0}$.
I don't see the need for this footnote: above we already saw that the disk lies in each of the domains $\Omega_{n_0+1},\Omega_{n_0+2}, \dots$, so clearly it lies within some domain of the $\Omega_n$'s.
What am I doing wrong here? Thanks.
EDIT: actually, I can't see how it that remark correct. Even if there is a finite subcovering of the disk $\{\Omega_{n_k} \}_{k=1}^N$, why should it lie in a single disk $\Omega_{n_0}$?
You are right: the footnote is both unnecessary and confusing.
Unnecessary because in the theorem we assume $f_n\to f$ uniformly on every compact subset of $\Omega$. This (implicitly) contains the assumption that every compact subset of $\Omega$ lies within the domain of $f_n$ for all sufficiently large $n$.
Confusing because "it is contained in a single $\Omega_{n_0}$" is not what the proof needs; we need the disk to be contained in $\Omega_n$ for all $n\ge n_0$.
Also confusing because "it is contained in a single $\Omega_{n_0}$" does not follow from the definition of a compact set, as the remark appears to suggest.
Verdict: disregard the footnote, continue reading the proof.