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.
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.
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.