Vitali's convergence theorem

Assume \(f _ { n } \in H ( \Omega )\) is a sequence of holomorphic functions on the region Ω that are uniformly bounded on compact subsets and \(f \in H ( \Omega )\) is such that the set \(\{ z \in \Omega : \lim _ { n \to \infty } f _ { n } ( z ) = f ( z ) \}\) has a limit point in Ω. Show that \(f _ { n }\) converges to f uniformly on compact subsets of Ω.