Problem 7. Fix a ∈ R and let S = {x ∈ R | x < a} = (−∞, a)….

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