Understanding Analysis
This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly fo
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Stephen Abbott
Understanding Analysis Second Edition
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA
Advisory Board: Colin Adams, Williams College David A. Cox, Amherst College Pamela Gorkin, Bucknell University Roger E. Howe, Yale University Michael Orrison, Harvey Mudd College Jill Pipher, Brown University Fadil Santosa, University of Minnesota
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an appreciation of interrelations among different aspects of the subject. They feature examples that illustrate key concepts as well as exercises that strengthen understanding.
More information about this series at http://www.springer.com/series/666
Stephen Abbott
Understanding Analysis Second Edition
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Stephen Abbott Department of Mathematics Middlebury College Middlebury, VT, USA
ISSN 0172-6056 ISSN 2197-5604 (electronic) Undergraduate Texts in Mathematics ISBN 978-1-4939-2711-1 ISBN 978-1-4939-2712-8 (eBook) DOI 10.1007/978-1-4939-2712-8 Library of Congress Control Number: 2015937969 Mathematics Subject Classification (2010): 26-01 Springer New York Heidelberg Dordrecht London © Springer Science+Business Media New York 2001, 2015 (Corrected at 2nd printing 2016) This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Science+Business Media LLC New York is part of Springer Science+Business Media (www. springer.com)
Preface My primary goal in writing Understanding Analysis was to create an elementary one-semester book that exposes students to the rich rewards inherent in taking a mathematically rigorous approach to the study of fu
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