Prove or disprove the following:
Let $\{g_\nu\}_{\nu\in\mathbb{N}}$ be a sequence of uniformly continuous functions over a compact set $X\subseteq\mathbb{R}^n$. Let $g$ be uniformly continuous over $X$ and assume that $g_\nu\to g$ point-wise. Then, $g_\nu\to g$ locally uniformly.
Can the same be assumed if the assumption on $g$ is dropped? i.e., does a point-wise convergent sequence of uniformly continuous functions, converge to its limit locally uniformly?
