Proposition 15

*If $X$ is a locally compact $\sigma$-compact space, there is a sequence $(U_n)$ of relatively compact open subsets of $X$ which cover $X$, such that $\overline{U}_n \subset U_{n+1}$ for each $n$.

Let $X$ be the union of a sequence $(K_n)$ of compact sets. Let $U_1$ be a relatively compact open neighbourhood of $K_1$ (no. 7, Proposition 10) and define $U_n$ inductively for $n > 1$ to be a relatively compact open neighbourhood of $\overline{U}_{n-1} \in K_n$ (no. 3, Proposition 5; no. 7, Proposition 10). The sequence $(U_n)$ clearly has the required properties.