An Introduction to REAL ANALYSIS An Introduction to REAL ANALYSIS Ravi P. Agarwal Cristina Flaut Donal O’Regan CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20171215 International Standard Book Number-13: 978-0-8153-9685-7 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Dedicated to our mothers: y Godawari Agarwal y Elena Paiu and Maria Paiu y Eileen O’Regan Contents Preface xi 1 Logic and Proof Techniques 1 2 Sets and Functions 11 3 Real Numbers 19 4 Open and Closed Sets 29 5 Cardinality 37 6 Real-Valued Functions 45 7 Real Sequences 53 8 Real Sequences (Contd.) 63 9 In(cid:12)nite Series 73 10 In(cid:12)nite Series (Contd.) 83 11 Limits of Functions 89 12 Continuous Functions 99 vii viii Contents 13 Discontinuous Functions 111 14 Uniform and Absolute Continuities and Functions of Bounded Variation 119 15 Differentiable Functions 129 16 Higher Order Differentiable Functions 141 17 Convex Functions 151 18 Indeterminate Forms 159 19 Riemann Integration 167 20 Properties of the Riemann Integral 177 21 Improper Integrals 189 22 Riemann-Lebesgue Theorem 197 23 Riemann-Stieltjes Integral 203 24 Sequences of Functions 211 25 Sequences of Functions (Contd.) 217 26 Series of Functions 229 27 Power and Taylor Series 237 Contents ix 28 Power and Taylor Series (Contd.) 243 29 Metric Spaces 251 30 Metric Spaces (Contd.) 257 Bibliography 267 Index 269