Problem 7. Fix a ˆˆ R and let S = {x ˆˆ R | x < a} = (ˆ’ˆž, a). Show that sup S = a. Use the density theorem to conclude that there is s ˆˆ S such that a ˆ’ ε < s < a. Problem 8. Consider a nonempty set S Š‚ R. Prove that S has a supremum if and only if the set ˆ’S = {x ˆˆ R | ˆ’x ˆˆ S} has an infimum, in which case inf(ˆ’E) = ˆ’ sup E.