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 is a bounded function on . If denote the least upper bounds and denote the greatest lower bounds of respectively, prove that .

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 is a bounded and integrable function on , and if and are the least upper and greatest lower bounds of over , then

if , and

if .

Also, since is a bounded and integrable function on , then is also bounded and integrable over .

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.