Show a normal family $\{f_n\} $ converges uniformly on compacts Let $\{f_n\}$ be a sequence of holomorphic functions on a domain $\Omega\subset \mathbb C$ which is bounded uniformly on compact subsets of $\Omega$. Let $\{z_k\}$ be a sequence of distinct points in $\Omega$.
with lim$_{k\to \infty} z_k = z_0 \in \Omega$. Assume that lim$_{n\to\infty} f_n(z_k)$ exists, for all $k$. Prove that
$\{f_n\}$ converges uniformly on compact subsets of  $\Omega$.
So far I can show that for disk $D$ centered at $z_k$, $\{f_n\}$ converges uniformly on $D$ if  $\{f_n^{(i)}(z_k)\}$ converges for all $i$. But I don't know how this will help in proving uniform convergence on arbitrary compacts. Any help will be appreciated.

It suffices to show that there exists a holomorphic $f$ with the property that every subsequence of $\{f_n\}$ has a further subsequence which converges to $f$: if this were to hold but $f_n\not\to f$, then by Montel’s theorem we can pass to a subsequence which converges to something that is not $f$, however this is impossible since passing to yet another subsequence shows that this “not $f$” is indeed $f$.
Now by Montel’s theorem yet again, every subsequence of $\{f_n\}$ has a further subsequence which converges to something; but by our condition on $\{z_k\}$, we know that all these convergent subsequences must agree: this is a consequence of the identity principle (the set of points where they agree has an accumulation point).
