Proof. Let $A, B \subset X$ be disjoint closed sets. By Theorem 5.20.10, there exists a partition of unity $\seqi{f}$ subordinate to $\bracs{A^c, B^c}$. Let $I = I_{A} \sqcup I_{B}$ such that for each $i \in I_{A}$, $\supp{f_i}\subset B^{c}$, and for each $i \in I_{B}$, $\supp{f_i}\subset A^{c}$. Take $f = \sum_{i \in I_A}f_{i}$ and $g = \sum_{i \in I_B}f_{i}$, then since $f|_{B} = 0$ and $g|_{A} = 0$, $\bracs{f \ge 2/3}\in \cn_{X}(A)$ and $\bracs{f \le 1/3}\in \cn_{X}(B)$.$\square$