Undergraduate Texts in Mathematics - Springer

Undergraduate Texts in Mathematics

Editorial Board S. Axler

K.A. Ribet

For other titles Published in this series, go to series/666

Matthias Beck ? Ross Geoghegan

The Art of Proof

Basic Training for Deeper Mathematics

Matthias Beck Department of Mathematics San Francisco State University San Francisco, CA 94132 USA beck@math.sfsu.edu

Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@sfsu.edu

Ross Geoghegan Department of Mathematical Sciences Binghamton University State University of New York Binghamton, NY 13902 USA ross@math.binghamton.edu

K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley.edu

ISSN 0172-6056

ISBN 978-1-4419-7022-0

e-ISBN 978-1-4419-7023-7

DOI 10.1007/978-1-4419-7023-7

Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010934105 Mathematics Subject Classification (2010): 00A05, 00A35

? Matthias Beck and Ross Geoghegan 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 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 ()

Great teachers introduced us to the arts of mathematics and writing:

To Harald Kohl and Hartmut Stapf

To the memory of Fr. Harry Lawlor, SJ and Fr. Joseph Veale, SJ

Preface

PEANUTS: c United Feature Syndicate, Inc. Reprinted with permission.

We have written this book with several kinds of readers in mind: (a) Undergraduates who have taken courses such as calculus and linear algebra,

but who are not yet prepared for upper-level mathematics courses. We cover mathematical topics that these students should know. The book also provides a bridge to the upper-level courses, since we discuss formalities and conventions in detail, including the axiomatic method and how to deal with proofs. (b) Mathematics teachers and teachers-in-training. We present here some of the foundations of mathematics that anyone teaching mathematics beyond the most elementary levels should know. (c) High-school students with an unusually strong interest in mathematics. Such students should find this book interesting and (we hope) unconventional. (d) Scientists and social scientists who have found that the mathematics they studied as undergraduates is not sufficient for their present needs. Typically, the problem here is not the absence of training in a particular technique, but rather a general feeling of insecurity about what is correct or incorrect in mathematics, a sense of material only partly understood. Scientists must be confident that they are using mathematics correctly: fallacious mathematics almost guarantees bad science. In so far as possible we try to "work in" the formal methods indirectly, as we take the reader through some interesting mathematics. Our subject is number systems: the integers and the natural numbers (that's the discrete Part I), the real numbers and the rational numbers (the continuous Part II). In this there is emphasis on induction,

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