The Real Numbers and Real Analysis
This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. The choice of material and the flexible organization, including three diffe
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Ethan D. Bloch
The Real Numbers and Real Analysis
Ethan D. Bloch Mathematics Department Bard College Annandale-on-Hudson, NY 12504 USA [email protected]
ISBN 978-0-387-72176-7 e-ISBN 978-0-387-72177-4 DOI 10.1007/978-0-387-72177-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011928556 Mathematics Subject Classification (2010): 26-01 © Springer Science+Business Media, LLC 2011 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 storage and 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 on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Dedicated to my two wonderful children Gil Nehemya and Ada Haviva, for whom my love has no upper bound
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
To the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi To the Instructor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxvii 1
Construction of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Entry 1: Axioms for the Natural Numbers . . . . . . . . . . . . . . . . . . . . . . 1.3 Constructing the Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Entry 2: Axioms for the Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Constructing the Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Dedekind Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Constructing the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Properties of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 Entry 3: Axioms for the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.3 Algebraic Properties of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . 65 2.4 Fin
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