# 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.