How would you prove this question: Let A and B be nonempty sets of the real numbers. Define A-B={a-b:a in A,b in B}. Show that if A and B are bounded, then sup(A-B)=supA - infB and inf(A-B)=infA - supB. Thanks a lot for any help.

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- Sep 22nd 2007, 09:24 AMBrainManleast upper bound and greatest lower bound
How would you prove this question: Let A and B be nonempty sets of the real numbers. Define A-B={a-b:a in A,b in B}. Show that if A and B are bounded, then sup(A-B)=supA - infB and inf(A-B)=infA - supB. Thanks a lot for any help.

- Sep 22nd 2007, 10:52 AMred_dog
Let .

Let

Let , so is a majorant for .

Let . Then .

Exist and such that and .

Then , so is not a majorant.

So, is the supremum for .

Proove in the same way the second part. - Sep 22nd 2007, 10:59 AMPlato
Suppose that then .

Now this means that

So is an upper bound for the set .

Now show that is the least upper bound.

.

From which we see that .

Thus no number less than is an upper bound for making the least upper bound for .

You need to fill many details. But this is the basic idea.