How to elegantly prove that one can always choose a closed chart within an open one on a manifold?

Question

Let $C \subset U \subset M$ with C compact and U open. For each $p \epsilon C$, choose a chart (x, V) with $\overline{V} \subset U$ and $x(p)=0$ This is just a bit from Spivak Vol1, p. 33-34. What is the tersest or most elegant way to prove that we can always choose such a V? The only way I can think of doing it is by using open rectangles in a chart and then using sub-rectangles but it would be kind of clunky. Is there a quicker purely topological way of doing it? The $x(p)=0$ part I don't really care about, but it makes me think that maybe I actually have to use a chart. Just to make clear, the claim is obvious, but the tersest rigorous proof is not.