Let $f_n$ be a locally bounded sequence of functions $f_n:G\to \mathbb{C}$. That is, for each $g\in G$, 1;;there exists an $r>0$ such that $f_n$ is globally bounded in $B_r(g)$. Suppose furthermore that $f_n$ converges pointwise on a dense subset $J\subseteq G$. Then $f_n\to f$ locally uniformly on $G$.

Proof:

• Use local Cauchy Criterion
• show locally bounded implies locally equi-lipschitz
• Use compactness of $\overline{B_r(a)}$