Let $f_n\colon \mathbb{C} \to \mathbb{C}$ be a sequence of entire functions, such that $f_n$ converges to the zero function on an open dense subset $U$ of $\mathbb{C}$ pointwise (or equivalently normally). Then, does $f_n$ genuinely converge to the zero function on $\mathbb{C}$?

It seems to me that it does converge, but I am not sure. If the assumption is a convergence on a dense subset of $\mathbb{C}$,then $f_n$ does not necessarily converge (https://math.stackexchange.com/questions/3651442/pointwise-convergence-of-holomorphic-functions-on-a-dense-set/3651462#3651462).

I would appreciate any comments! Thanks.