ABSTRACT
This chapter introduces the related concept of a Cauchy sequence. In a similarly imprecise way, it says that a sequence is Cauchy if the terms of that sequence get “closer and closer” to each other. To determine whether or not a Cauchy sequence converges, one need to make some more connections. The first comes from the following lemma: If a sequence is Cauchy, then it is bounded. To proceed, one need to introduce the notion of a subsequence, which can be thought of as an infinite subset of the original sequence (with the relative ordering remaining unchanged). One might ask whether the convergence of a sequence implies convergence of its subsequences, and (conversely) whether subsequence convergence implies sequence convergence. The chapter presents complete proofs of all lemmas, theorems, and corollaries. It uses the scaffolding provided to complete a proof of the Monotone Subsequence Lemma.
