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.



Convergence of a sequence of entire functions on an open dense subset
It is not true. Consider the compact sets
$$K\_n=\{ z:|z|\leq n, |\arg z|\geq 1/n\}\cup\{0\}.$$
By Runge's approximation theorem, there exist polynomials $f\_n$,
such that $|f\_n(z)|<1/n,\; z\in K\_n,$ and $f\_n(1)=n.$
This sequence of polynomials evidently converges uniformly on compact subsets of the dense open set $C\backslash[0,+\infty)$ but does not converge at the point $1$.


