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 is integrable, so is the absolute-value function . Prove this by showing that if refer to , and refer to , then . Then use the following theorem.

(for this, is a lower sum, and is an upper sum)

"Suppose is bounded on , and suppose that corresponding to each positive there is a partition of such that the corresponding upper and lower sums satisfy the inequality . Then is integrable."

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