Let $\{f_n\}n\in \mathbb{N}$ be a l1;;ocally bounded sequence of holomorphic functions $f_n:G\to \mathbb{C}$. Then it posesses a locally uniformly convergent subsequence.

PROOF:

• Pick a countable dense subset of $J\subseteq G$
• Consider pointwise-convergence at points of $J$.
• Apply a argument like Cantors' diagonal argument to get a sequence converging pointwise on $J$.
• Apply the 1;;previous lemma