# Convergence of a sequence of entire functions on an open dense subset

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.

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 $$1$$.
Due to Weierstrass there exists an entire function $$F$$ such that: $$F(0)\neq 0$$ and $$F(re^{it})\to 0$$ if $$r$$ tends to $$\infty$$, for any real $$t$$. Then $$g_n(z):=F(nz)$$ gives a counterexample. See the book by Remmert "Funktionentheorie II", chapter 12, §3. I am sure that there is an English translation.