Introduction to Real Analysis

This is a text for a two-term course in introductory real analysis for junior or senior mathematics majors and engineering and science students with a serious interest in mathematics. Prospective educators of mathematically gifted high school students can also benefit from the mathematical maturity that can be gained from an introductory real analysis course. The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics. The standard elementary calculus sequence is the only specific prerequisite for Chapters 1-5, which deal with real-valued functions. (However, other analysis oriented courses, such as elementary differential equations, provide useful preparatory experience.) Chapters 6 and 7 require a working knowledge of determinants, matrices, and linear transformations, typically available from a first course in linear algebra. Chapter 8 is accessible after completion of Chapters 1-5. Without taking a position for or against current reforms in mathematics teaching, I think it is fair to say that the transition from elementary courses such as calculus, linear algebra, and differential equations to a rigorous real analysis course is a bigger step today than it was just a few years ago. To make this step, today's students need more help than their predecessors did, and must be coached and encouraged more. Therefore, while striving throughout to maintain a high level of rigor, I have tried to write as clearly and informally as possible. In this connection, I find it useful to address the student in the second person. I have included 295 completely worked out examples to illustrate and clarify all major theorems and definitions. I have emphasized careful statements of definitions and theorems and have tried to be complete and detailed in proofs, except for omissions left to the exercises. I give a thorough treatment of real-valued functions before considering vector-valued functions. In making the transition from one to several variables and from real-valued to vector-valued functions, I have left to the student some proofs that are essentially repetitions of proofs of earlier theorems. I believe that working through the details of straightforward generalizations of more elementary results is good practice for the student. Great care has gone into the preparation of the 760 numbered exercises, many with multiple parts. These range from routine to very difficult. Hints are provided for the more difficult parts of the exercises. ORGANIZATION Chapter 1 is concerned with the real number system. Section 1.1 begins with a brief discussion of the axioms for a complete ordered field, but no attempt is made to develop the reals from them; rather, it is assumed that the student is familiar with the consequences of these axioms, except for one: the completeness axiom. Since the difference between a rigorous and a nonrigorous treatment of calculus can be described largely in terms of the attitude taken toward completeness, I have expended considerable effort in developing its consequences. Section 1.2 is about mathematical induction. Although this may seem out of place in a real analysis course, I have found that the typical beginning real analysis student simply cannot do an induction proof without reviewing the method. Section 1.3 is devoted to elementary set theory and the topology of the real line, ending with the Heine-Borel and Bolzano-Weierstrass theorems. Chapter 2 covers the differential calculus of functions of one variable: limits, continuity, differentiability, 1'Hospital's rule, and Taylor's theorem. The emphasis is on rigorous presentation of principles; no attempt is made to develop the properties of specific elementary functions. Even though this may not be done rigorously in most contemporary calculusTrench, William F. is the author of 'Introduction to Real Analysis', published 2002 under ISBN 9780130457868 and ISBN 0130457868.