I know a few things about how I'd go about this question, though I don't have all the pieces I need. This question probably isn't that hard.

Suppose that $\displaystyle f(x)$ is a bounded function on $\displaystyle [a,b]$. If $\displaystyle M, M'$ denote the least upper bounds and $\displaystyle m, m'$ denote the greatest lower bounds of $\displaystyle f, |f|$ respectively, prove that $\displaystyle M'-m'\leq M-m$.

The only materials I have right now is fairly simple stuff relating to this sort of thing. It's a pair of theories (for which I don't have the proofs on hand).

If $\displaystyle f$ is a bounded and integrable function on $\displaystyle [a,b]$, and if $\displaystyle M$ and $\displaystyle m$ are the least upper and greatest lower bounds of $\displaystyle f$ over $\displaystyle [a,b]$, then

$\displaystyle m(b-a)\leq \int_a^b f(x)dx\leq M(b-a)$ if $\displaystyle a\leq b$, and

$\displaystyle m(b-a)\geq \int_a^b f(x)dx\geq M(b-a)$ if $\displaystyle b\leq a$.

Also, since $\displaystyle f$ is a bounded and integrable function on $\displaystyle [a,b]$, then $\displaystyle |f|$ is also bounded and integrable over $\displaystyle [a,b]$.

Those are probably most of what would be needed to solve this, though I'm having some difficulties putting the pieces together. If I could get a hand with it, that'd be great.