Elementary Real Analysis

PREFACE University mathematics departments have for many years offered courses with titles such asAdvanced CalculusorIntroductory Real Analysis.These courses are taken by a variety of students, serve a number of purposes, and are written at various levels of sophistication. The students range from ones who have just completed a course in elementary calculus to beginning graduate students in mathematics. The purposes are multifold: To present familiar concepts from calculus at a more rigorous level. To introduce concepts that are not studied in elementary calculus but that are needed in more advanced undergraduate courses. This would include such topics as point set theory, uniform continuity of functions, and uniform convergence of sequences of functions. To provide students with a level of mathematical sophistication that will prepare them for graduate work in mathematical analysis, or for graduate work in several applied fields such as engineering or economics. To develop many of the topics that the authors feel all students of mathematics should know. There are now many texts that address some or all of these objectives. These books range from ones that do little more than address objective (1) to ones that try to address all four objectives. The books of the first extreme are generally aimed at one-term courses for students with minimal background. Books at the other extreme often contain substantially more material than can be covered in a one-year course. The level of rigor varies considerably from one book to another, as does the style of presentation. Some books endeavor to give a very efficient streamlined development; others try to be more user friendly. We have opted for the user-friendly approach. We feel this approach makes the concepts more meaningful to the student. Our experience with students at various levels has shown that most students have difficulties when topics that are entirely new to them first appear. For some students that might occur almost immediately when rigorous proofs are required. For others, the difficulties begin with elementary point set theory, compactness arguments, and the like. To help students with the transition from elementary calculus to a more rigorous course, we have included motivation for concepts most students have not seen before and provided more details in proofs when we introduce new methods. In addition, we have tried to give students ample opportunity to see the new tools in action. For example, students often feel uneasy when they first encounter the various compactness arguments (Heine-Borel theorem, Bolzano-Weierstrass theorem, Cousin's lemma, introduced in Section 4.5). To help the student see why such theorems are useful, we pose the problem of determining circumstances under which local boundedness of a functionfon a setEimplies global boundedness offonE.We show by example that some conditions onEare needed, namely thatEbe closed and bounded, and then show how each of several theorems could be used to show that closed and boundedness of the setEsuffices. Thus we introduce students to the theorems by showing how the theorems can be used in natural ways to solve a problem. We have also included some optional material, marked as "Advanced" or "Enrichment" and flagged with a scissors symbol. Enrichment We have indicated as "Enrichment"' some relatively elementary material that could be added to a longer course to provide enrichment and additional examples. For example, in Chapter 3 we have added to the study of series a section on infinite products. While such a topic plays an important role in the representation of analytic functions, it is presented here to allow the instructor to explore ideas that are closely related to the study of series and that help illustrate and review many of the fundamentaThomson, Brian S. is the author of 'Elementary Real Analysis', published 2000 under ISBN 9780130190758 and ISBN 0130190756.