Real Analysis Problems

Problem Books in Mathematics
Edited by P. Winkler

For other titles in this series, go to http://www.springer.com/series/714

Asuman G. Aksoy Mohamed A. Khamsi

A Problem Book in Real Analysis

123

Asuman G. Aksoy Department of Mathematics Claremont McKenna College Claremont, CA 91711 USA aaksoy@cmc.edu Series Editor: Peter Winkler Department of Mathematics Dartmouth CollegeHanover, NH 03755 USA peter.winkler@dartmouth.edu

Mohamed A. Khamsi Department of Mathematical Sciences University of Texas at El Paso El Paso, TX 79968 USA mohamed@utep.edu

ISSN 0941-3502 ISBN 978-1-4419-1295-4 e-ISBN 978-1-4419-1296-1 DOI 10.1007/978-1-4419-1296-1 Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2009939759 Mathematics Subject Classification(2000): 00A07 c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storageand retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed onacid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Dedicated to Erc¨ment G. Aksoy and Anny Morrobel-Sosa u

Contents
Preface 1 Elementary Logic and Set Theory Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 9

2 Real Numbers 21 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 27 3 Sequences 41 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Limits of Functions 63 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5 Continuity 77 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6Differentiability 97 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7 Integration 127 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8 Series 159 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9 Metric Spaces 181 Solutions . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 10 Fundamentals of Topology 197 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 223 11 Sequences and Series of Functions Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Bibliography Index 249 251

viiPreface
Education is an admirable thing, but it is well to remember from time to time that nothing worth knowing can be taught. Oscar Wilde, “The Critic as Artist,” 1890.

Analysis is a profound subject; it is neither easy to understand nor summarize. However, Real Analysis can be discovered by solving problems. This book aims to give independent students the opportunity to discover RealAnalysis by themselves through problem solving. The depth and complexity of the theory of Analysis can be appreciated by taking a glimpse at its developmental history. Although Analysis was conceived in the 17th century during the Scientific Revolution, it has taken nearly two hundred years to establish its theoretical basis. Kepler, Galileo, Descartes, Fermat, Newton and Leibniz were among those who...