I think that if the sequence $f_n$ converges pointwise to $f$ (and not uniformly), $f$ is not always holomorphic. Then I searched concrete examples of $f_n$ such that $f$ is not holomorphic, but I couldn't find such a $f_n$. As a side note, I thought of $U$ as a unit disk $D\colon=\{z\in\mathbb{C}\mid |z|<1\}$ when I searched that.
Is there the concrete example? I would appreciate it if you propose that example!