Let \(K \subset \mathbb{R}^{d}\) be a compact set. Show that there is a decreasing sequence of continuous functions \(\phi_{n}(x)\) such that \(\mathbb{1}_{K}=\inf _{n} \phi_{n}\).

Let \(U \supset K\) be an open set and \(\phi(x):=d\left(x, U^{c}ight) /\left(d(x, K)+d\left(x, U^{c}ight)ight)\).