Peculiar means of proving a theorem on integrability

This theorem has probably been seen a dozen times here, and I could find other formats of proving this theorem online, but the issue I'm having right now is a context/format issue. The question, listed below, is very specific on how we're supposed to answer it, and I'm not entirely sure how I'm supposed to format it.

If $\displaystyle f$ is integrable, so is the absolute-value function $\displaystyle |f(x)|$. Prove this by showing that if $\displaystyle s,S$ refer to $\displaystyle f$, and $\displaystyle s',S'$ refer to $\displaystyle |f|$, then $\displaystyle S'-s'\leq S-s$. Then use the following theorem.

(for this, $\displaystyle s$ is a lower sum, and $\displaystyle S$ is an upper sum)

"Suppose $\displaystyle f$ is bounded on $\displaystyle [a,b]$, and suppose that corresponding to each positive $\displaystyle \epsilon$ there is a partition of $\displaystyle [a,b]$ such that the corresponding upper and lower sums satisfy the inequality $\displaystyle S-s<\epsilon$. Then $\displaystyle f$ is integrable."

I'm guessing that to solve this one, I'd first need to solve this earlier question: click here.