Walter Rudin Principles Of Mathematical Analysis McGraw Hill 1976 
PRINCIPLES OF
MATHEM ATICAL
ANALYSIS
McGR1\W-HILL BOOK COMPANY
Auckland Bogota Guatemala Hamburg Lisbon
London Madrid Mexico Ne\v Delhi Panama Paris
Sao Pau lo Singapore Sydney Tokyo
WALTER RUDIN
Professor of Mathematics
University of Wisconsin-Madison
Principles of
Mathematical
Analysis
THIRD EDITION
San
Juan
PRINCIPLES
OF MATHEMATICAL ANALYSIS, Third Edition
International Editions 1976
Exclusive rights by McGraw-Hill Book Co. - Singapore for manufacture and export. This book cannot be re-exported from the country to which it is consigned by McGraw-Hill.
Copyright © 1 964, 1 976 by McGraw-Hill, Inc. All rights reserved.
Copyright © 1 953 by McGraw-Hill, Inc. All rights reserved.
Except as pennitted under the United States Copyright Act of 1 976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.
9 3 30 KKP 2 0 9
Library of Congress Cataloging-in-Publication Data
Rudin, Walter, date
Principles of mathematical analysis.
(International series in pure and applied mathematics)
Bibliography: p.
Includes index.
1 . Mathematical analysis
I. Title.
QA300.R8
1 976
ISBN 0-07-054234-X
51 5
75- 17903
When orderibg this title, use ISBN 0-07-085613-3
Printed in Singapore
CONTENTS
.
Preface
Chapter 1
lX
The Real and Complex Number Systems
1
Introduction
I
Ordered Sets
3
Fields
5
The Real Field
8
The Extended Real N un1bcr Sy�ten1
The Con1plex Field
Euclidean Spaces
Appendix
Chap ter 2
II
I2
I6
17
Exercises
2I
Basic Topology
24
Finite, Countable, and Uncountable Sets
24
30
36
41
Metric Spaces
Compact Sets
Perfect
Sets
vi
CONTE NTS
Chapter 3
Connected Sets
Exercises
42
Numerical Sequences and Series
47
Convergent Sequences
Subsequences
Cauchy Sequences
Upper and Lower Li mits
Some Special Sequences
Series
Series of Nonnegative Terms
The N u mber e
The Root and Ratio Tests
Power Series
MATHEM ATICAL
ANALYSIS
McGR1\W-HILL BOOK COMPANY
Auckland Bogota Guatemala Hamburg Lisbon
London Madrid Mexico Ne\v Delhi Panama Paris
Sao Pau lo Singapore Sydney Tokyo
WALTER RUDIN
Professor of Mathematics
University of Wisconsin-Madison
Principles of
Mathematical
Analysis
THIRD EDITION
San
Juan
PRINCIPLES
OF MATHEMATICAL ANALYSIS, Third Edition
International Editions 1976
Exclusive rights by McGraw-Hill Book Co. - Singapore for manufacture and export. This book cannot be re-exported from the country to which it is consigned by McGraw-Hill.
Copyright © 1 964, 1 976 by McGraw-Hill, Inc. All rights reserved.
Copyright © 1 953 by McGraw-Hill, Inc. All rights reserved.
Except as pennitted under the United States Copyright Act of 1 976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.
9 3 30 KKP 2 0 9
Library of Congress Cataloging-in-Publication Data
Rudin, Walter, date
Principles of mathematical analysis.
(International series in pure and applied mathematics)
Bibliography: p.
Includes index.
1 . Mathematical analysis
I. Title.
QA300.R8
1 976
ISBN 0-07-054234-X
51 5
75- 17903
When orderibg this title, use ISBN 0-07-085613-3
Printed in Singapore
CONTENTS
.
Preface
Chapter 1
lX
The Real and Complex Number Systems
1
Introduction
I
Ordered Sets
3
Fields
5
The Real Field
8
The Extended Real N un1bcr Sy�ten1
The Con1plex Field
Euclidean Spaces
Appendix
Chap ter 2
II
I2
I6
17
Exercises
2I
Basic Topology
24
Finite, Countable, and Uncountable Sets
24
30
36
41
Metric Spaces
Compact Sets
Perfect
Sets
vi
CONTE NTS
Chapter 3
Connected Sets
Exercises
42
Numerical Sequences and Series
47
Convergent Sequences
Subsequences
Cauchy Sequences
Upper and Lower Li mits
Some Special Sequences
Series
Series of Nonnegative Terms
The N u mber e
The Root and Ratio Tests
Power Series