Infinite Sequences III
In Theorem 4.10, we proved that for a sequence to converge, a necessary condition is the boundedness of the sequence, and in our example of the sequence (−1) n , we saw that boundedness is not a sufficient condition for convergence.
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Miklós Laczkovich Vera T. Sós
Real Analysis
Foundations and Functions of One Variable
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 Lisette G. de Pillis, Harvey Mudd College Jill Pipher, Brown University Fadil Santosa, University of Minnesota
Undergraduate Texts in Mathematics are generally aimed at third- and fourthyear 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
Mikl´os Laczkovich
•
Vera T. S´os
Real Analysis Foundations and Functions of One Variable Fifth Edition
123
Mikl´os Laczkovich Department of Analysis E¨otv¨os Lor´and University Budapest, Hungary
Vera T. S´os Alfr´ed R´enyi Institute of Mathematics Hungarian Academy of Sciences Budapest, Hungary
ISSN 0172-6056 ISSN 2197-5604 (electronic) Undergraduate Texts in Mathematics ISBN 978-1-4939-2765-4 ISBN 978-1-4939-2766-1 (eBook) DOI 10.1007/978-1-4939-2766-1 Library of Congress Control Number: 2015938228 Springer New York Heidelberg Dordrecht London 1st edition: T. S´os, Vera, Anal´ızis I/1 © Nemzeti Tank¨onyvkiad´o, Budapest, 1972 2nd edition: T. S´os, Vera, Anal´ızis A/2 © Nemzeti Tank¨onyvkiad´o, Budapest, 1976 3rd edition: Laczkovich, Mikl´os & T. S´os, Vera: Anal´ızis I © Nemzeti Tank¨onyvkiad´o, Budapest, 2005 4th edition: Laczkovich, Mikl´os & T. S´os, Vera: Anal´ızis I © Typotex, Budapest, 2012 Translation from the Hungarian language 3rd edition: Val´os anal´ızis I by Mikl´os Laczkovich & T. S´os, Vera, © Nemzeti Tank¨onyvkiad´o, Budapest, 2005. All rights reserved © Springer 2015. © Springer New York 2015 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 editor
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