This page intentionally left blank Real Analysis PURE AND APPLIED MATHEMATICS A Wiley Series of Texts, Monographs, and Tracts Founded by RICHARD COURANT Editors Emeriti: MYRON B. ALLEN III, DAVID A. COX, PETER HILTON, HARRY HOCHSTADT, PETER LAX, JOHN TOLAND A complete list of the titles in this series appears at the end of this volume. Real Analysis: A Historical Approach Second Edition Saul Stahl The University of Kansas Department of Mathematics Lawrence, KS )WILEY A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representation or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Stahl, Saul. Real analysis : a historical approach / Saul Stahl. — 2nd ed. p. cm. Includes index. ISBN 978-0-470-87890-3 (hardback) 1. Mathematical analysis. 2. Functions of real variables. I. Title. QA300.S882 2011 515'.8—dc22 2011010976 Printed in the United States of America. oBook ISBN: 978118096864 ePDF ISBN: 978118096840 ePub ISBN: 978118096857 10 9 8 7 6 54 3 2 1 This book is dedicated to the memory of my parents and my brother, Finkla, Moses, and Dan Stahl This page intentionally left blank Contents Preface to the Second Edition xi Acknowledgments xv 1 Archimedes and the Parabola 1 1.1 The Area of the Parabolic Segment 1 1.2* The Geometry of the Parabola 8 2 Fermat, Differentiation, and Integration 13 2.1 Fermat's Calculus 13 3 Newton's Calculus (Part 1) 19 3.1 The Fractional Binomial Theorem 19 3.2 Areas and Infinite Series 23 3.3 Newton's Proofs 30 4 Newton's Calculus (Part 2) 35 4-1 The Solution of Differential Equations 35 4-2* The Solution of Algebraic Equations 40 Chapter Appendix: Mathematica Implementations of Newton's Algorithm 48 VII νιϊι CONTENTS 5 Euler 51 5.1 Trigonometrie Series 51 6 The Real Numbers 61 6.1 An Informal Introduction 61 6.2 Ordered Fields 64 6.3 Completeness and Irrational Numbers 71 6.4* The Euclidean Process 76 6.5 Functions 80 7 Sequences and Their Limits 85 7.1 The Definitions 85 7.2 Limit Theorems 92 8 The Cauchy Property 103 8.1 Limits of Monotone Sequences 103 8.2 The Cauchy Property 111 9 The Convergence of Infinite Series 115 9.1 Stock Series 115 9.2 Series of Positive Terms 120 9.3 Series of Arbitrary Terms 124 9.4* The Most Celebrated Problem 132 10 Series of Functions 139 10.1 Power Series 139 10.2 Trigonometric Series 145 11 Continuity 149 11.1 An Informal Introduction 149 11.2 The Limit of a Function 150 11.3 Continuity 155 11.4 Properties of Continuous Functions 163 12 Differentiability 169 12.1 An Informal Introduction to Differentiation 169 12.2 The Derivative 171 12.3 The Consequences of Differentiability 177