Proposition 7.2.5. Let $(X, d)$ be a metric space, $C \subset X$ be closed, and $K \subset X$ be compact. If $C \cap K = \emptyset$, then $d(K, C) > 0$.
Proof. Suppose that $d(K, C) = 0$, then there exists $\seq{(x_n, y_n)}\subset K \times C$ such that $d(x_{n}, y_{n}) \to 0$ as $n \to \infty$. By Definition 4.16.1, there exists a subsequence $\seq{n_k}$ and $x \in K$ such that $x_{n_k}\to x$ as $k \to \infty$. In which case, $d(x, C) = 0$, and $x \in \ol{K}$ by Proposition 7.2.4, so $K \cap C \ne \emptyset$.$\square$