Advanced Calculus : An Introduction To Mathematical Analysis

yXbVancel ywvancec Calculus XiS /B \ * 7 ^ ^ s* ^m >^ > An Introduction to Mathematical Analysis Mathematical Analysis This page is intentionally left blank i)s,w^ meol AfrJancel L7 n^ * * , . Calculus lalculus An Introduction to Mathematical Analysis 5. Zailman University of Montreal World Scientific Singapore'NewJerseyLondorrHongKong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Zaidman, Samuel, 1933Advanced calculus : an introduction to mathematical analysis / S. Zaidman. p. cm. Includes bibliographical references (p. 171) and indexes. ISBN 9810227043 1. Mathematical analysis. I. Title. QA300.Z285 1997 515»dc21 97-20207 CIP British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 1997 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. This book is printed on acid-free paper. Printed in Singapore by Uto-Print PREFACE The present book is, as its title indicates, a presentation of some fundamental ideas related to the elementary real analysis. For a better understanding of the material given here, a course in basic differential and integral calculus can be a good preliminary preparation. The emphasis in our present text lies on the so-called rigorous method where everything (well, almost everything) is given a clear definition, a detailed statement, a complete, logically coherent proof. The book starts with an exposition of the theory of real numbers; it is this theory which is taken as a basis for the concepts developed afterwards. We present real numbers as equivalence classes of Cauchy sequences of rational numbers. This is different from what can be found in most recent books, where one prefers the axiomatic way or (sometimes) the method of "Dedekmd cuts." We consider the method of Cauchy sequences a very clear manner of introducing general, real numbers; furthermore, it has the advantage of being applicable in other situations, for instance in the theory of "metric spaces." Afterwards we give the usual, modern presentation, of such topics as: sequences of real numbers, infinite numerical series, continuous functions, derivatives and integration theory. There are also two chapters of a peculiar type: the first one concerns convex functions, a class of functions which appear useful in many applications of calculus; the second one, about metric spaces, reflects a recent tendency to start presenting prest;ii uxii£, "topological" l u p u i u ^ i ^ d i ideas from the very beginning of the undergraduate mathematical life. The attentive reader will note probably the absence of most, "well-known" elementary functions, like sin x, cos rr, ex, log x, from the book. They are V vi Preface never used here; their ri