First formulation(I don't understand this one and don't know where to find the proof):
Let $D$ be a domain.
Let $\{f_n(z)\}$ be a sequence of functions such that for all $n\in\mathbb{N}$:
(i) $f_n(z)$ is holomorphic for all $z\in D$;
(ii) $f_n(z)\neq a$, $f_n(z)\neq b$ for all $z\in D$, where $a\neq b$ are distinct complex numbers.
Let $\Delta$ be an infinite set with at least one limit point in $D$.
Assume that $\{f_n(z)\}$ converges to a finite value for all $z\in \Delta$.
Then $\{f_n(z)\}$ converges uniformly in each compact subset of $D$ to a function holomorphic in $D$.
Second formulation (I understand this one):
Let $D$ be a simply connected open domain.
Let $\{f_n(z)\}$ be a sequence of functions such that for all $n\in\mathbb{N}$:
(i) $f_n(z)$ is holomorphic for all $z\in D$;
(ii) $f_n(z)$ is uniformly bounded over any compact subdomain of $D$.
Let $\Delta$ be an infinite set with at least one limit point in $D$.
Assume that $\{f_n(z)\}$ converges to a finite value for all $z\in \Delta$.
Then $\{f_n(z)\}$ converges uniformly in each compact subdomain of $D$ to a function holomorphic in $D$.
Can anybody explain the condition (ii) in the first formulation, i.e. why this condition is needed and what it actually means? Is it possible to obtain one formulation of the theorem from another?
My thoughts: I think condition (ii) in the 1st formulation might be related to the Great Picard's theorem https://en.wikipedia.org/wiki/Picard_theorem :
If an analytic function $f$ has an essential singularity at a point $w$, then on any punctured neighborhood of $w$, $f(z)$ takes on all possible complex values, with at most a single exception, infinitely often.
Is this so?
I totally understand the second formulation and its proof. But the first formulation is complete mystery to me. I know the first formulation is related to normal family of functions studied by Montel, but i know next to nothing about such abstract notions. First formulation is supposed to be on pages 248-251 of the book Analytic function theory, volume 2 by Hille, but I couldn't find it there, although there is something about system of equations $f_n(z)= a$, $f_n(z)= b$, $f_n(z)= c$, but nothing about only two points. There is also theorem 15.3.1 on page 251, but it again has nothing about two points. This only further confused me. Slogging through all the definitions and theorems in Hille's book to understand what is a normal family of functions and Montel's work seems almost impossible to me either, especially when the first formulation does not seem to be in this book anyway. Hope someone can shed light on this question and answer it without invoking complicated abstract notions beyond the standard courses of complex analysis. Thanks.
You are right that the first formulation is a strong result that is related with Picard's little theorem (in the sense that Picard's little theorem is a corollary of its methods of proof) and with a little more work it can be shown that it implies Picard's great theorem too; it follows fairly easily from Schottky's Theorem which implies that if $\mathscr F$ is a family of analytic functions on some disc that omit two (finite) values and are uniformly bounded at the center of the disc, then the family is locally uniformly bounded.
An easy connectedness argument extends this to any family of analytic functions on some domain that omit two finite complex values and are uniformly bounded at a point in the domain.
So with the hypothesis of the "First formulation", the sequence $f_n$ is locally uniformly bounded in $D$ hence it is normal, so any subsequence has a normally convergent subsequence but the limit is unique by the identity theorem, so the family converges normally