Lectures on Real Analysis

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1 Lectures on Real Analysis This is a rigorous introduction to real analysis for undergraduate students, starting from the axioms for a complete ordered field and a little set theory. The book avoids any preconceptions about the real numbers and takes them to be nothing but the elements of a complete ordered field. All of the standard topics are included, as well as a proper treatment of the trigonometric functions, which many authors take for granted. The final chapters of the book provide a gentle, example-based introduction to metric spaces with an application to differential equations on the real line. The author s exposition is concise and to the point, helping students focus on the essentials. Over 200 exercises of varying difficulty are included, many of them adding to the theory in the text. The book is ideal for second-year undergraduates and for more advanced students who need a foundation in real analysis. in this web service

2 AUSTRALIAN MATHEMATICAL SOCIETY LECTURE SERIES Editor-in-chief: Professor C. Praeger, School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia Editors: Professor P. Broadbridge, School of Engineering and Mathematical Sciences, La Trobe University, Victoria 3086, Australia Professor Michael Murray, School of Mathematical Sciences, University of Adelaide, SA 5005, Australia Professor C. E. M. Pearce, School of Mathematical Sciences, University of Adelaide, SA 5005, Australia Professor M. Wand, School of Mathematical Sciences, University of Technology, Sydney, NSW 2007, Australia 1 Introduction to Linear and Convex Programming, N. CAMERON 2 Manifolds and Mechanics, A. JONES, A. GRAY & R. HUTTON 3 Introduction to the Analysis of Metric Spaces, J. R. GILES 4 An Introduction to Mathematical Physiology and Biology, J. MAZUMDAR 5 2-Knots and their Groups, J. HILLMAN 6 The Mathematics of Projectiles in Sport, N. DE MESTRE 7 The Petersen Graph, D. A. HOLTON & J. SHEEHAN 8 Low Rank Representations and Graphs for Sporadic Groups, C. E. PRAEGER & L. H. SOICHER 9 Algebraic Groups and Lie Groups, G. I. LEHRER (ed.) 10 Modelling with Differential and Difference Equations, G. FULFORD, P. FORRESTER & A. JONES 11 Geometric Analysis and Lie Theory in Mathematics and Physics, A. L. CAREY & M. K. MURRAY (eds.) 12 Foundations of Convex Geometry, W. A. COPPEL 13 Introduction to the Analysis of Normed Linear Spaces, J. R. GILES 14 Integral: An Easy Approach after Kurzweil and Henstock, L. P. YEE & R. VYBORNY 15 Geometric Approaches to Differential Equations, P. J. VASSILIOU & I. G. LISLE (eds.) 16 Industrial Mathematics, G. R. FULFORD & P. BROADBRIDGE 17 A Course in Modern Analysis and its Applications, G. COHEN 18 Chaos: A Mathematical Introduction, J. BANKS, V. DRAGAN & A. JONES 19 Quantum Groups, R. STREET 20 Unitary Reflection Groups, G. I. LEHRER & D. E. TAYLOR in this web service

3 Australian Mathematical Society Lecture Series: 21 Lectures on Real Analysis FINNUR LÁRUSSON University of Adelaide in this web service

4 cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by, New York Information on this title: / C 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of. First published 2012 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Lárusson, Finnur, 1966 Lectures on real analysis /. pages cm. (Australian Mathematical Society lecture series ; 21) ISBN (hardback) 1. Mathematical analysis. I. Title. QA300.5.L dc ISBN Hardback ISBN Paperback has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. in this web service

5 Contents Preface To the student vii ix Chapter 1. Numbers, sets, and functions The natural numbers, integers, and rational numbers Sets Functions 9 More exercises 11 Chapter 2. The real numbers The complete ordered field of real numbers Consequences of completeness Countable and uncountable sets 19 More exercises 21 Chapter 3. Sequences Convergent sequences New limits from old Monotone sequences Series Subsequences and Cauchy sequences 32 More exercises 35 Chapter 4. Open, closed, and compact sets Open and closed sets 39 v in this web service

6 vi Contents 4.2. Compact sets 41 More exercises 42 Chapter 5. Continuity Limits of functions Continuous functions Continuous functions on compact sets and intervals Monotone functions 51 More exercises 53 Chapter 6. Differentiation Differentiable functions The mean value theorem 59 More exercises 60 Chapter 7. Integration The Riemann integral The fundamental theorem of calculus The natural logarithm and the exponential function 69 More exercises 71 Chapter 8. Sequences and series of functions Pointwise and uniform convergence Power series Taylor series The trigonometric functions 83 More exercises 87 Chapter 9. Metric spaces Examples of metric spaces Convergence and completeness in metric spaces 95 More exercises 99 Chapter 10. The contraction principle The contraction principle Picard s theorem 107 More exercises 111 Index 113 in this web service

7 Preface This book is a rigorous introduction to real analysis, suitable for a onesemester course at the second-year undergraduate level, based on my experience of teaching this material many times in Australia and Canada. My aim is to give a treatment that is brisk and concise, but also reasonably complete and as rigorous as is practicable, starting from the axioms for a complete ordered field and a little set theory. Along with epsilons and deltas, I emphasise the alternative language of neighbourhoods, which is geometric and intuitive and provides an introduction to topological ideas. I have included a proper treatment of the trigonometric functions. They are sophisticated objects, not to be taken for granted. This topic is an instructive application of the theory of power series and other earlier parts of the book. Also, it involves the concept of a group, which most students won t have seen in the context of analysis before. There may be some novelty in the gentle, example-based introduction to metric spaces at the end of the book, emphasising how straightforward the generalisation of many fundamental notions from the real line to metric spaces really is. The goal is to develop just enough metric space theory to be able to prove Picard s theorem, showing how a detour through some abstract territory can contribute back to analysis on the real line. Needless to say, I claim no originality whatsoever for the material in this book. My contribution, such as it is, lies in the selection and presentation of the material. I thank the American Mathematical Society for allowing the book to be formatted with one of their class files. vii in this web service

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9 To the student The purpose of this course is twofold. First, to give a careful treatment of calculus from first principles. In first-year calculus we learn methods for solving specific problems. We focus on how to use these methods more than why they work. To pave the way for further studies in pure and applied mathematics we need to deepen our understanding of why, as opposed to how, calculus works. This won t be a simple rehashing of first-year calculus at all. Calculus done this way is called real analysis. In particular, we will consider what it is about the real numbers that makes calculus work. Why can t we make do with the rationals? We will identify the key property of the real numbers, called completeness, that distinguishes them from the rationals and permeates all of mathematical analysis. Completeness will be our main theme through the whole course. The second goal of the course is to practise reading and writing mathematical proofs. The course is proof-oriented throughout, not to encourage pedantry, but because proof is the only way that mathematical truth can be known with certainty. Mathematical knowledge is accumulated through long chains of reasoning. We can t rely on this knowledge unless we re sure that every link in the chain is sound. In many future endeavours, you will find that being able to construct and communicate solid arguments is a very useful skill. With the emphasis on rigorous arguments comes the need to make our fundamental assumptions, from which our reasoning begins, clear and explicit. We shall list ten axioms that describe the real numbers and that can in fact be shown to characterise the real numbers. Our development of real analysis will be based on these axioms, along with a bit of set theory. ix in this web service

10 x To the student Towards the end of the course we extend some of the concepts we will have developed in the context of the real numbers to the much more general setting of metric spaces. To demonstrate the power of abstraction, the course ends with the proof, using metric space theory, of an existence and uniqueness theorem for solutions of differential equations. in this web service

Advanced Grammar in Use

Advanced Grammar in Use A self-study reference and practice book for advanced learners of English Third Edition with answers and CD-ROM cambridge university press cambridge, new york, melbourne, madrid,