LIBRARY OF MATHEMATICS edited by WALTER LEDERMANN D.Sc., Ph.D., F.R.S.Ed., Professor of Mathematics, University of Sussex Linear Equations Sequences and Series Differential Calculus Elementary Differential Equations and Operators Partial Derivatives Complex Numbers Principles of Dynamics Electrical and Mechanical Oscillations Vibrating Systems Vibrating Strings Fourier Series Solutions of Laplace's Equation Solid Geometry Numerical Approximation Integral Calculus Sets and Groups Differential Geometry Probability Theory Multiple Integrals Fourier and Laplace Transforms Introduction to Abstract Algebra Functions of a Complex Variable, 2 Vols Linear Programming Sets l!l1d Numbers P. M. Cohn J. A. Green P. J. Hilton G. E. H. Reuter P. J. Hilton W.Ledermann M. B. Glauert D. S. Jones R. F. Chisnell D. R. Bland I. N. Sneddon D. R. Bland P. M. Cohn B. R. Morton W. Ledermann J. A. Green K. L. Wardle A. M. Arthurs W. Ledermann P. D. Robinson C. R. J. Clapham D. O. Tall Kathleen Trustrum S. 8wierczkoswki
INTRODUCTION TO MATHEMATICAL ANALYSIS BY C. R. J. CLAPHAM Department of Mathematics University of Aberdeen ROUTLEDGE & KEGAN PAUL LONDON AND BOSTON
First published in 1973 by Routledge & Kegan Paul Ltd, Broadway House, 68-74 Carter Lane, London EC4V 5EL and 9 Park Street, Boston, Mass. 02108, U.S.A. Willmer Brothers Limited, Birkenhead C. R. J. Clapham, 1973 No part of this book may be reproduced in any form without permission from the publisher, except for the quotation of brief passages in criticism ISBN-13: 978-0-7100-7529-1 e-isbn-13: 978-94-011-6572-3 DOl: 10.1007/978-94-011-6572-3 Library of Congress Catalog Card No. 72-95122
Contents Preface page vii 1. Axioms for the Real Numbers 1 Introduction 1 2 Fields 1 3 Order 6 4 Completeness 9 5 Upper bound 11 6 The Archimedean property 13 Exercises 15 2. Sequences 7 Limit of a sequence 18 8 Sequences without limits 22 9 Monotone sequences 23 Exercises 25 3. Series 10 Infinite series 27 11 Convergence 27 12 Tests 29 13 Absolute convergence 31 14 Power series 32 Exercises 34
4. Continuous Functions 15 Limit of a function 36 16 Continuity 37 17 The intermediate value property 40 18 Bounds of a continuous function 41 Exercises 43 5. Differentiable Functions 19 Derivatives 45 20 Rolle's theorem 47 21 The mean value theorem 49 Exercises 51 6. The Riemann Integral 22 Introduction 54 23 Upper and lower sums 56 24 Riemann-integrable functions 57 25 Examples 60 26 A necessary and sufficient condition 62 27 Monotone functions 63 28 Uniform continuity 65 29 Integrability of continuous functions 67 30 Properties of the Riemann integral 68 31 The mean value theorem 72 32 Integration and differentiation 73 Exercises 76 Answers to the Exercises 78 Index 81
Preface I have tried to provide an introduction, at an elementary level, to some of the important topics in real analysis, without avoiding reference to the central role which the completeness of the real numbers plays throughout. Many elementary textbooks are written on the assumption that an appeal to the completeness axiom is beyond their scope; my aim here has been to give an account of the development from axiomatic beginnings, without gaps, while keeping the treatment reasonably simple. Little previous knowledge is assumed, though it is likely that any reader will have had some experience of calculus. I hope that the book will give the non-specialist, who may have considerable facility in techniques, an appreciation of the foundations and rigorous framework of the mathematics that he uses in its applications; while, for the intending mathematician, it will be more of a beginner's book in preparation for more advanced study of analysis. I should finally like to record my thanks to Professor Ledermann for the suggestions and comments that he made after reading the first draft of the text. University of Aberdeen C. R. 1. CLAPHAM