Question on the proof of de Rham's theorem

Question

In John M. Lee's Introduction to Smooth Manifolds, in chapter 18, he seeks to prove the de Rham theorem. Step 4 of the proof is to show that if $M$ is a smooth manifold with a de Rham basis, then $M$ satisfies the de Rham theorem ($M$ is "de Rham"). In his proof of this claim he states: Let $\{U_\alpha\}$ is such a basis, and $f:M\to \mathbb R$ is a smooth exhaustion function for $M$, and define the following subsets of $M$: $$A_m := \{q\in M| m \le f(q) \le m+1\}$$$$A'_m:=\{q\in M|m -\frac 12 < f(q) < m+\frac32\}.$$ Then, for each $q\in M$, there is a basis open subset containing $q$ and contained in $A'_m$ such that all of these open subsets cover $A_m$. He offers no justification for these statements, and I have beeen unable to justify them myself.