INSTRUCTOR’S MANUAL TO ACCOMPANY
INTRODUCTION TO REAL
ANALYSIS
Fourth Edition
Robert G. Bartle
Eastern Michigan University
Donald R. Sherbert
University of Illinois
JOHN WILEY & SONS, INC.
New York • Chichester • Weinheim • Brisbane • Singapore • Toronto
,Copyright
c 2000, 2010 by John Wiley & Sons, Inc.
Excerpts from this work may be reproduced by instructors
for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses
for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that
permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright
owner is unlawful. Requests for permission or further
information should be addressed to the Permissions
Department, John Wiley & Sons, Inc., 111 River Street,
Hoboken, NJ 07030-5774, (201)-748-6011, Fax (201) 748-6008,
Website http://www.wiley.com/go/permissions.
ISBN 978-0-471-44799-3
,PREFACE
This manual is offered as an aid in using the fourth edition of Introduction to Real
Analysis as a text. Both of us have frequently taught courses from the earlier
editions of the text and we share here our experience and thoughts as to how to
use the book. We hope our comments will be useful.
We also provide partial solutions for almost all of the exercises in the book.
Complete solutions are almost never presented here, but we hope that enough is
given so that a complete solution is within reach. Of course, there is more than
one correct way to attack a problem, and you may find better proofs for some of
these exercises.
We also repeat the graphs that were given in the manual for the previous
editions, which were prepared for us by Professor Horacio Porta, whom we wish
to thank again.
Robert G. Bartle November 20, 2010
Donald R. Sherbert
, CONTENTS
Chapter 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 The Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 3 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
Chapter 5 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter 6 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
Chapter 7 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Chapter 8 Sequences of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter 9 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Chapter 10 The Generalized Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter 11 A Glimpse into Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Selected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
INTRODUCTION TO REAL
ANALYSIS
Fourth Edition
Robert G. Bartle
Eastern Michigan University
Donald R. Sherbert
University of Illinois
JOHN WILEY & SONS, INC.
New York • Chichester • Weinheim • Brisbane • Singapore • Toronto
,Copyright
c 2000, 2010 by John Wiley & Sons, Inc.
Excerpts from this work may be reproduced by instructors
for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses
for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that
permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright
owner is unlawful. Requests for permission or further
information should be addressed to the Permissions
Department, John Wiley & Sons, Inc., 111 River Street,
Hoboken, NJ 07030-5774, (201)-748-6011, Fax (201) 748-6008,
Website http://www.wiley.com/go/permissions.
ISBN 978-0-471-44799-3
,PREFACE
This manual is offered as an aid in using the fourth edition of Introduction to Real
Analysis as a text. Both of us have frequently taught courses from the earlier
editions of the text and we share here our experience and thoughts as to how to
use the book. We hope our comments will be useful.
We also provide partial solutions for almost all of the exercises in the book.
Complete solutions are almost never presented here, but we hope that enough is
given so that a complete solution is within reach. Of course, there is more than
one correct way to attack a problem, and you may find better proofs for some of
these exercises.
We also repeat the graphs that were given in the manual for the previous
editions, which were prepared for us by Professor Horacio Porta, whom we wish
to thank again.
Robert G. Bartle November 20, 2010
Donald R. Sherbert
, CONTENTS
Chapter 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 The Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 3 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
Chapter 5 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter 6 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
Chapter 7 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Chapter 8 Sequences of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter 9 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Chapter 10 The Generalized Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter 11 A Glimpse into Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Selected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
