Exhaustion of a Locally Compact Separable Metric Space by Compact Sets, with Cut-Off Functions

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $(X,d)$ be a locally compact metric space that is separable. There are compact sets $K_{m}\subseteq X$ and functions $\varphi_{m}$ in $C_{c}(X)$, for $m\in\mathbb{N}$, with the following properties.

(i) Exhaustion. For every $m\in\mathbb{N}$, $K_{m}\subseteq\operatorname{int}K_{m+1}\subseteq K_{m+1}$; and

$$\bigcup_{m\in\mathbb{N}}K_{m}=X.$$

(ii) Compact sets. For every compact $C\subseteq X$ there is $m\in\mathbb{N}$ with $C\subseteq\operatorname{int}K_{m}$.

(iii) Cut-off functions. For every $m\in\mathbb{N}$, $0\le\varphi_{m}(x)\le1$ for every $x\in X$, $\varphi_{m}(x)=1$ for every $x\in K_{m}$, and the support of $\varphi_{m}$ satisfies $\operatorname{supp}\varphi_{m}\subseteq K_{m+1}$.