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Undergraduate Texts in Mathematics
S. Axler
F.W. Gehring
K.A. Ribet
New York
Hong Kong
Ulrich Daepp Pamela Gorkin
Reading, Writing,
and Proving
A Closer Look at Mathematics
Ulrich Daepp Pamela Gorkin
Department of Mathematics Department of Mathematics
Bucknell University
Bucknell University
Lewisburg, PA 17837
Lewisburg, PA 17837
Editorial Board:
S. Axler
F.W. Gehring
K.A. Ribet
Mathematics Department Mathematics Department
Mathematics Department
San Francisco State
East Hall
University of California,
University of Michigan
San Francisco, CA 94132
Ann Arbor, MI 48109
Berkeley, CA 94720-3840
Mathematics Subject Classification (2000): 00-01, 03-01
Library of Congress Cataloging-in-Publication Data
Daepp, Ulrich.
Reading, writing, and proving : a closer look at mathematics / Ulrich Daepp, Pamela
p. cm.– (Undergraduate texts in mathematics)
Includes bibliographical references and index.
ISBN 0-387-00834-9 (alk. paper)
1. Mathematics–Study and teaching (Higher)–United States. 2. Technical
writing–Study and teaching (Higher)–United States. I. Gorkin, Pamela. II. Title.
III. Series.
QA113.D34 2003
ISBN 0-387-00834-9
Printed on acid-free paper.
© 2003 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without
the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue,
New York, NY 10010, 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 in the United States of America.
SPIN 10920411
Springer-Verlag New York Berlin Heidelberg
A member of BertelsmannSpringer Science + Business Media GmbH
For Hannes and Madeleine
You are probably about to teach or take a “first course in proof
techniques,” or maybe you just want to learn more about mathemat-
ics. No matter what the reason, a student who wishes to learn the
material in this book likes mathematics, and we hope to keep it that
way. At this point, students have an intuitive sense of why things
are true, but not the exposure to the detailed and critical thinking
necessary to survive in the mathematical world. We have written
this book to bridge this gap.
In our experience, students beginning this course have little
training in rigorous mathematical reasoning; they need guidance.
At the end, they are where they should be; on their own. Our aim
is to teach the students to read, write, and do mathematics inde-
pendently, and to do it with clarity, precision, and care. If we can
maintain the enthusiasm they have for the subject, or even create
some along the way, our book has done what it was intended to do.
Reading . This book was written for a course we teach to first and
second year college students. The style is informal. A few problems
require calculus, but these are identified as such. Students will also
need to participate while reading proofs, prodded by questions (such
as, “Why?”). Many detailed examples are provided in each chapter.
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