$\Omega$ is a domain of holomorphy, $D=\{z \in C : |z|\leq1\}$.For an arbitrary series of holomorphic functions $F_k:D\to \Omega$,The closure of $\cup_{k\geq1}F_k(\partial D)$ is compact in $\Omega$. Show that the closure of ${\cup_{k\geq1}F_k(D)}$ is also compact in $\Omega$.
I have tried to consider this problem from the definition of a domain of holomorphy. However, I did not solve it.
Actually, I do not have any useful idea about this question.
So sorry for being a beginning learner, and thanks in advance.