Uniform limit of analytic continuation.

Question

Let $V \subset U \subset \mathbb{C}$ be two connected open sets.

Let us assume to have a sequence of functions $f_n$ which are defined and holomorphic on $V$ and that, for each $n$, $f_n$ has an analytical continuation $\tilde{f}_n$ on $U$. Assume that $f_n \to f$ uniformly on $V$. Hence $f$ is holomorphic on $V$.

Does there exists $\tilde{f}$ holomorphic on $U$ such that $\tilde{f}_n \to \tilde{f}$ on $U$ ?

Answer 1

Hint: Consider $V=D(0,1/2),U=D(0,3/2), f_n(z) = z^n.$