A Concrete Approach to Classical Analysis
Mathematical analysis offers a solid basis for many achievements in applied mathematics and discrete mathematics. This new textbook is focused on differential and integral calculus, and includes a wealth of useful and relevant examples, exercises, and res
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Marian Mures¸an
A Concrete Approach to Classical Analysis
Canadian Mathematical Society Société mathématique du Canada
Canadian Mathematical Society Ouvrages de math´ematiques de la SMC Editor s-in-Chief ´ Redacteurs-en-chef Jonathan Borwein Peter Borwein
Marian Mure s¸ an
A Concrete Approach to Classical Analysis
ABC
Marian Mures¸an Faculty of Mathematics & Computer Science Babes¸-Bolyai University Str. Mihail Kogalniceanu nr. 1 400084 Cluj-Napoca Romania [email protected]
ISBN 978-0-387-78932-3 e-ISBN 978-0-387-78933-0 DOI 10.1007/978-0-387-78933-0 Library of Congress Control Number: 2008939066 Mathematics Subject Classification (2000): 46-xx c Springer Science+Business Media, LLC 2009 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 9 8 7 6 5 4 3 2 1 springer.com
To the memory of my mother
vii
Computo ergo sum D. H. Bailey 1
´ o˜ιδα o´τ ι δ´ν o˜ιδα Eν (The only thing I know is that I don’t know anything) Socrates 2
Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about. Old Testament, 1 Kings 7:23
We would be amiss, however, if we did not emphasize that the extended precision calculation of π has substantial application as a test of the “global integrity” of a supercomputer. D. H. Bailey, J. M. Borwein,3 and P. B. Borwein4 in [10]
. . . the main goal of computation in pure mathematics is arguably to yield insight. [22]
1 2 3 4
David H. Bailey, Chief Technologist, Computational Research Dept., Lawrence Berkeley Laboratory. Socrates of Athens (Σωκρατ ης o Aϑηναιoς) ∼ 470– ∼ 399 (BC). Jonathan Michael Borwein, 1951– . Peter Benjamin Borwein, 1953– .
Contents
1
Sets and Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The concept of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Relations and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Sets of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Two examples . . . . . . . . . . . . . . . . . .
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