Let S ⊂ R be nonempty. Prove that if a number u in R has the properties: (i) for every n ∈ N the number u - 1/n is not an upper bound of S, and (ii) for every number n ∈ N the number u + 1/n is an upper bound of S, then u = sup S. (This is the converse of Exercise 2.3.9.)