I am learning about lebesgue integrals on my own using Apostol, and am stuck on an exercise. Let $\{r_1,r_2,...\}$ denote the set of rational numbers in [0,1] and let $I_n=(r_n-4^{-n},r_n+4^{-n}) \cap I$. Let $f(x)$ =1 if $x \in I_n$ for some $n$ and let $f(x)=0$ otherwise. The end goal of this exercise is to show that $f$ is an upper function, but $-f$ is not. I am stuck on part (c) of this exercise:
If a step function $s$ satisfies $s(x) \leq -f(x)$ on I, show that $s(x) \leq -1$ almost everywhere on $I$.
I tried to show this by showing $f(x) \geq 1$ almost everywhere, but I am not convinced that's true anymore. Any suggestions on how I should proceed with this problem would be greatly appreciated.
HINT: Suppose $E_1 = \{x \in I : s(x) > -1\}$ has positive measure. Show there exists some $x \in E_1$ such that $s$ is constant on some interval $J$ containing $x$. Then note that $J$ contains some rational number $q$, but $s(q) \leq -1$.