the question states if a and b are bounded above what can on

the question states if a and b are bounded above what can one say about sup (a) and sup (b)

Solution

By definition of sup(S), s sup(S) for all s S.  

Now we also know that if sup(A) is greater than a number x, then some element of A is greater than x.

Suppose that sup(A) > sup(B). Then there exists a A such that a > sup(B).

Thus we have found an element of A which cannot lie in B, contradicting that A B.

Hence if a and b are bounded above and A B we can say that sup(A) sup(B).