Suppose $E\subset \mathbb{R}^d$ is a given set, $m$ is the Lebesgue measure, and $\mathcal{O}_n$ is the open set:
$\mathcal{O}_n = \{x : d(x, E) < 1/n\}.$
The goal is for me to show that if $E$ is compact, then $m(E) = \lim\limits_{n \to \infty} m(\mathcal{O}_n)$.
I am having trouble not only visualizing these sets, but also intuitively realizing what this means. In other words, I have no idea how to begin this proof.