Let $n\in\mathbb{N}$ and let $\{f_j\}_{j=1}^\infty$ be a sequence of Lipschitz functions $f_j:\mathbb{R}^n\to\mathbb{R}^n$ with uniform Lipschitz constant $L\in(0,\infty)$. Suppose there is a function $f:\mathbb{R}^n\to\mathbb{R}^n$ (not necessarily Lipschitz or even continuous) for which $f_j\to f$ pointwise, i.e. $\lim_{j\to\infty}f_j(x)=f(x)$ for all $x\in\mathbb{R}^n$. Does it follow that $f_j\to f$ uniformly, i.e. that $\lim_{j\to\infty}\sup_{x\in\mathbb{R}^n}|f_j(x)-f(x)|=0$?
I know this is true for Lipschitz sequences on compact metric spaces, and that it is not true for arbitrary metric spaces. However, I seem to recall hearing that it was true for $\mathbb{R}$, which would make it true for $\mathbb{R}^n$ too, I believe.
Thanks!