Generalizing Montel's theorem to functions omtting a value from each of 3 disjoint compact sets. I want to prove the following.

Let $K_1,K_2,K_3$ be three disjoint compact subsets of the Riemann sphere, and let $f_1, f_2,...$ be meromorphic functions on a domain $D$ such that each $f_n$ omits at least one point from each of the sets $K_i$. Then there is a subsequence $f_{n_k}$ that converges normally on all compact subsets of $D$.

The convergence is in the spherical metric. The case where each of the $K_i$ is a single point is the well-known Montel's fundamental normality test
. 
When each of $K_i$ is a finite set we can deduce the claim directly from Montel, since there is only a finite combination of possibilites for what values are omitted from each set, so there is a subsequence that always omits the same 3 points and we can apply the usual Montel's theorem. For more general compact subsets such an argument breaks down.
To prove the claim, it is enough to show that on every compact subset $K$ there is a bound $M_K$ on the spherical derivatives $\frac{2|f'(z)|}{1+|f(z)|^2}$.
However, I believe I need to somehow reduce again to the case of finite $K_i$, since the proof of the usual Montel's theorem is very tricky. It proceeds by assuming WLOG that the domain is the unit disk $\Bbb D$ (which can be done here) and then that the omitted values are $0,1,\infty$ (which has no obvious analog in our case), looking for every $k$ at the $2^k$th roots $g_n$ of the functions $f_n$, assuming by contradiction that they are not normal and showing that the functions given by Zalcman's lemma converge normally to a function that is shown to be constant.
 A: W.L.O.G we may assume that $\infty \not \in K_1\cup K_2\cup K_3$.
Let us first consider the case when $\{f_n\}\subseteq \textbf{Hol}(\Omega)$, and each $f_n$ misses at least one point from each of the compact sets $K_1$ and $K_2$. Choose $a_n\in K_1\setminus \textbf{Im}(f_n)$ and $b_n\in K_2\setminus \textbf{Im}(f_n)$. As $K_1$ and $K_2$ are compact, $\exists$ convergent subsequences of $\{a_n\}$ and $\{b_n\}$, say $\{a_{n_k}\}$ and $\{b_{n_k}\}$ resp. Now define 
$$g_{n_k}=\dfrac{f_{n_k}-a_{n_k}}{b_{n_k}-a_{n_k}}$$
Consider the family $\{g_{n_k}\}$. It omits $0$ and $1$, and therefore is normal. I'll leave it to you to check that the normality of $\{f_{n_k}\}$ follows.
The meromorphic version follows rather directly after this. Let $\{f_n\}\subseteq\textbf{Mer}(\Omega)$, where each $f_n$ misses at least one point from each of the compact sets $K_1$, $K_2$ and $K_3$. Let $f_n$ omit $c_n\in K_3$. Choose a convergent subsequence of $\{c_n\}$, say $\{c_{n_k}\}$. Define
$$g_{n_k}=\dfrac{1}{f_{n_k}-c_{n_k}}$$
Consider the family $\{g_{n_k}\}\subseteq \textbf{Hol}(\Omega)$. It is normal by the previous case (Why ?). Now it follows that on any compact set $K \subset \Omega$,
$$\dfrac{|f'_{n_k}|}{1+|f_{n_k}-c_{n_k}|^2}=\dfrac{|g'_{n_k}|}{1+|g_{n_k}|^2}\leq M_K$$
which by Marty's Theorem, implies that the family $\{f_{n_k}-c_{n_k}\}$ is normal. It is now easy to see that the family $\{f_{n_k}\}$ is normal, hence, the assertion. 
