Let $K_1 \subsetneq K_2$ be two non-empty compact sets and let $D = (d_n)_{n \in \mathbb{N}}$ be a dense sequence on $K_2\smallsetminus K_1.$ Consider $f_n : \mathbb{C}\smallsetminus K_1 \rightarrow \mathbb{C}$ to be a sequence of analytic functions. Assume there exists an analytic function $f : \mathbb{C} \smallsetminus K_2 \rightarrow \mathbb{C}$ such that $f_n$ converges uniformly on compact sets of $\mathbb{C} \smallsetminus K_2$ to $f$ and that $f_n$ converges pointwisely on the dense set $D$, but $f_n$ does not converge for the rest of points of its domain.

I would like to construct such an example or to prove that this is impossible, or at least to get a better understanding of this situation: it seems quite a strange behaviour for a sequence of analytic functions but I am not able to get a contradiction. If you drop some assumptions yo can get both results: if $K_1$ is empty then $f_n$ are entire functions and you can apply the maximum modulus principle to show that $f_n$ must converges uniformly on the whole complex plane. If you just ask $f_n$ to converges pointwisely on some dense sequence, you can get an infinite product with zeros on the sequence such that diverges on the rest of $\mathbb{C}.$

Can anyone help me?

Thank you very much.