Assume we have sets $\Omega_1 \subsetneq \Omega \subset \mathbb{C}$, both open and connected. Further, let $f_n$ and $f$ be holomorphic functions on $\Omega$ such that $f_n \to f$ uniformly on each compact subset of $\Omega_1$. Can we conclude anything about convergence on $\Omega$?
I have no intuition about the answer. Would anyone provide an idea for a proof or counterexample?
No. If $f_n(z)=1+z+z^2+\cdots+z^n$, then $(f_n)_{n\in\mathbb N}$ converges uniformly to $\frac1{1-z}$ on every compact subset of the open disc $D(0,1)$. However, $D(0,1)$ is the largest open subset $\Omega$ of $\mathbb C$ such that $(f_n)_{n\in\mathbb N}$ converges uniformly on every compact subset of $\Omega$.