let A,B be nonempty sets of real numbers, prove that: if A,B are bounded and they are disjoint, then supA doesnt equal supB. here's my proof: assume that supA=supB=c then for every a in A a<=c and for every b in B b<=c. bacuse A.B are bounded then: for every e>0 there exists x in A such that c-e<x<=c and there exists y in B such that c-e<y<=c so we have two elements that are both in A and B, but this is a contradiction. is this proof valid? i feel that i should show that y=x, but i think bacuse e is as we choose, we have to find elements which are both in A and B.