TAILIEUCHUNG - Ebook Problems in Real Analysis (Second Edition): Part 1

(BQ) Problems in Real Analysis is the ideal companion for senior science and engineering undergraduates and first-year graduate courses in real analysis. It is intended for use as an independent source, and is an invaluable tool for students who wish to develop a deep understanding and proficiency in the use of integration methods. | PROBLEMS IN . REAL ANALYSIS Second Edition A Workbook with Solutions CHARALAMBOS D. ALIPRANTIS Departments of Economics and Mathematics Purdue University and OWEN BURKINSHAW Departments of Mathematical Sciences Indiana University Purdue University Indianapolis ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto pk ỵ ữ Ằ 63 ms This book is printed on acid-free paper. @ Copyright 1999 1990 by Academic Press AH Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopy recording or any information storage and retrieval system without permission in writing from the publisher. ACADEMIC PRESS a division of Harcourt Brace Company 525 B Street Suite 1900 San Diego CA 92101-4495 USA http ACADEMIC PRESS LIMITED 24-28 Oval Road London NW1 7DX UK http ap International Standard Book Number 0-12-050253-4 This book accompanies the following title catalogued by the Library of Congress Library of Congress Cataloging-in-Publication Data Aỉiprantís Charaỉambos D. Problems in real analysis Charaỉambos D. Aliprantis and Owen Burkinshaw. p. cm. Includes bibliographical references and index. ISBN 0-12-050257-7 acid-free paper 1. Mathematical analysis. 2. Functions of real variables. Ĩ. Burkmshaw Owen. n. Title. 1998 515 dc21 . 98-3955 CĨP Printed in the United States of America 98 99 00 01 02 DS 9 8 7 6 5 4 3 2 1 CONTENTS Foreword vii CHAPTER 1. FUNDAMENTALS OF REAL ANALYSIS 1 1. Elementary Set Theory 1 2. Countable and Uncountable Sets 6 3. The Real Numbers 11 4. Sequences of Real Numbers 20 5. The Extended Real Numbers 34 6. Metric Spaces 45 7. Compactness in Metric Spaces 54 CHAPTER 2. TOPOLOGY AND CONTINUITY 65 8. Topological Spaces 65 9. Continuous Real-Valued Functions 73 10. Separation Properties of Continuous Functions 92 11. The Stone-Weierstrass Approximation Theorem 98 CHAPTER 3. THE THEORY OF MEASURE 107 12. Semirings and Algebras

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