Complex Analysis – Sequence of holomorphic functions converging on compact sets with $ f_n (D) subset U $ implies $ f (D) subset U $ or $ f (z) = c $ for $ c in partial U $

Let $ D $ to be an open connected set and $ {f_n } $ to be a sequence of holomorphic functions on $ D $ that converge uniformly to a holomorphic function $ f $ on all the compact subsets of $ D $. Assume that $ f_n (D) subset U $ for everyone $ n $. Show that $ f (D) subassembly U $ or there is a $ c in partial U $ such as $ f (z) = c $ for everyone $ z in D $.

First I assumed that $ f $ it was not a constant function. So I guess there's a $ w in D $ such as $ f (w) notin overline {U} $ By the open mapping theorem, I can take a little ball around this point. Consider closing this ball. Then removing this bullet under $ f $ is a closed set in $ D $. I then hope to use uniform convergence on compact sets to show that $ n $ wide enough $ f_n (D) not subset U $ which would be a contradiction.

I do not feel confident with this approach. In addition, this assignment has been given in a set of problems that uses the Schwarz reflection principle as well as the Caratheodory theorem. So maybe I should use this theorem.