Chapter A Preliminaries of Real Analysis A principal objective of this largely rudimentary chapter is to introduce the basic set-theoretical nomenclature that we adopt throughout the text. We start with an intuitive discussion of the notion of “set,” and then introduce the basic operations on sets, Cartesian products, and binary relations. | Chapter A Preliminaries of Real Analysis A principal objective of this largely rudimentary chapter is to introduce the basic set-theoretical nomenclature that we adopt throughout the text. We start with an intuitive discussion of the notion of set and then introduce the basic operations on sets Cartesian products and binary relations. After a quick excursion to order theory in which the only relatively advanced topic that we cover is the completion of a partial order functions are introduced as special cases of binary relations and sequences as special cases of functions. Our coverage of abstract set theory concludes with a brief discussion of the Axiom of Choice and the proof of Sziplrajn s Theorem on the completion of a partial order. We assume here that the reader is familiar with the elementary properties of the real numbers and thus provide only a heuristic discussion of the basic number systems. No construction for the integers is given in particular. After a short elaboration on ordered fields and the Completeness Axiom we note without proof that the rational numbers form an ordered field and real numbers a complete ordered field. The related discussion is intended to be read more quickly than anywhere else in the text. We next turn to real sequences. These we discuss relatively thoroughly because of the important role they play in real analysis. In particular even though our coverage will serve only as a review for most readers we study here the monotonic sequences and subsequential limits with some care and prove a few useful results like the Bolzano-Weierstrass Theorem and Dirichlet s Rearrangement Theorem. These results will be used freely in the remainder of the text. The final section of the chapter is nothing more than a swift refresher on the analysis of real functions. First we recall some basic definitions and then very quickly go over the concepts of limits and continuity of real functions defined on the real line. We then review the elementary .
