Given nonempty subsets A and B of positive real numbers, define Show that sup(A

Question

Given nonempty subsets A and B of positive real numbers, define Show that sup(A B) = (sup A)(sup B). Show that Show that, if A and B are bounded subsets of real numbers, then sup (A - B) = max {(sup A)(sup B), (sup A)(inf B), (inf A)(sup B), (inf A)(inf B)}. Give an example of two nonempty bounded sets A and B for which sup(A B) (sup A)(sup B)

Explanation / Answer

1.If either of A or B is unbounded, then so is A+B, and we see that sup(A+B) = sup(A) + sup(B) = infinity (assuming extended real number arithmetic.) Hereafter, assume both A and B are bounded. ---------------------- Since A and B are bounded sets in R, for any a in A and for any b in B, there exist a1, a2 in A and b1, b2 in B such that a1 a>=? From this inequality, we have 1-??/?/?>1/a>=1/? 1/?-?/?>1/a>=1/? Set b = 1/a then b?B. Furthermore, we have from (2) that ?-? sup(A)-u. Thus there exists a in A U B such that a>max{sup(A),sup(B)}-u for any u>0. This tells us that max{sup(A),sup(B)} is the Least upper bound of A U B, or that sup(A U B) = max{sup(A),sup(B)}.

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