Let \((f_n)\) be a sequence of continuous functions on \(\mathbb{R}^n\) which is uniformly bounded, i.e. there is a number \(M>0\) such that
\begin{equation*} \lvert f_n(x)\rvert\le M. \end{equation*}for every function \(f_n\) and every \(x\in \mathbb{R}^n\). The sequence is also uniformly equicontinuous, i.e. for every \(\varepsilon>0\) there is a \(\delta>0\) such that
\begin{equation*} \lvert f_n(x)-f_n(y)\rvert<\varepsilon \end{equation*}for all functions \(f_n\) whenever \(\lvert x-y\rvert<\delta\). Then there exists a subsequence \((f_{n_k})\) and a continuous function \(f\), such that \(f_{n_k}\to f\) locally uniformly , that is the sequence converges uniformly on every compact subset of \(\mathbb{R}^n\).