If sup A< sup B, then show that there exists an element b ? B that is an upper b

Question

If sup A< sup B, then show that there exists an element b ? B that is an upper bound for A.

Explanation / Answer

--------->Given that sup(B) is the least upper bound this means that it is the smallest number that is an upper bound of the set B. ----------->That means given any epsilon_{0}, no matter how small, there exist b_{0}inB such that b_{0} > sup(B) -epsilon_{0}. ---------->Lets start again. ----------->sup(B) - sup(A) = lpha >0 ----------->If we pick any epsilon_{0} such that epsilon_{0} < lpha we can find an element in B such that b_{0} > sup(B) -epsilon. So b_{0} is an upper bound of A.

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