In this chapter, we prove Theorem 4.2. For the sake of efficiency, our proof will be essentially algebraic. However, a clear geometric picture underlies our constructions. We discussed this geometric picture in §19.2. The reader might want to reread that section before going through the proof here. Also, the reader might want to review the proof we gave of Lemma 4.3 in § 18.3. Our proof here is similar to the one given there.

Let*A*be any irrational parameter. Let*A*. Let*S*be a monotone subsequence of the superior sequence. We