Page 1 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS
SOLUTIONS
MANUAL ANALYSIS
MANUAL ANALYSIS
WITH AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOFTO
FIFTH
PROOF
EDITION
FIFTHSteven
EDITION
LaySteven Lay.pdf
VE
R
INSTRUCTOR’S
IF
SOLUTIONS MANUAL
IE
D
BR
A NALYSIS WITH AN
AI
N
I NTRODUCTION TO PROOF
BO
FIFTH EDITION
O
ST
ER
Steven Lay
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
VERIFIEDBRAINBOOSTER
-2026 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS MANUAL
SOLUTIONS
ANALYSIS
MANUAL
WITH
ANALYSIS
AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOF FIFTH
TO PROOF
EDITION
FIFTH
Steven
EDITION
Lay Steven Lay
,Page 2 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS
SOLUTIONS
MANUAL ANALYSIS
MANUAL ANALYSIS
WITH AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOFTO
FIFTH
PROOF
EDITION
FIFTHSteven
EDITION
LaySteven Lay.pdf
VE
R
IF
IE
D
BR
AI
N
BO
O
ST
ER
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages
in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2014, 2005, 2001 Pearson Education, Inc.
Publishing as Pearson, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-74748-8
ISBN-10: 0-321-74748-8
www.pearsonhighered.com
VERIFIEDBRAINBOOSTER
-2026 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS MANUAL
SOLUTIONS
ANALYSIS
MANUAL
WITH
ANALYSIS
AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOF FIFTH
TO PROOF
EDITION
FIFTH
Steven
EDITION
Lay Steven Lay
,Page 3 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS
SOLUTIONS
MANUAL ANALYSIS
MANUAL ANALYSIS
WITH AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOFTO
FIFTH
PROOF
EDITION
FIFTHSteven
EDITION
LaySteven Lay.pdf
VE
Instructor’s Solutions Manual
R
to accompany
IF
IE
D
BR
ANALYSIS
AI
with an Introduction
N
to Proof
BO
5th Edition
O
ST
ER
Steven R. Lay
Lee University
& RS\ULJKW3HDUVRQ( GXFDWLRQ,QF
Upper Saddle River, New Jersey 07458
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
-2026 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS MANUAL
SOLUTIONS
ANALYSIS
MANUAL
WITH
ANALYSIS
AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOF FIFTH
TO PROOF
EDITION
FIFTH
Steven
EDITION
Lay Steven Lay
, Page 4 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS
SOLUTIONS
MANUAL ANALYSIS
MANUAL ANALYSIS
WITH AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOFTO
FIFTH
PROOF
EDITION
FIFTHSteven
EDITION
LaySteven Lay.pdf
VE
2
R
IF
IE
This manual is intended to accompany the 5th edition of Analysis with an Introduction to Proof
by Steven R. Lay (Pearson, 2013). It contains solutions to nearly every exercise in the text. Those
D
exercises that have hints (or answers) in the back of the book are numbered in bold print, and the hints
BR
are included here for reference. While many of the proofs have been given in full detail, some of the
more routine proofs are only outlines. For some of the problems, other approaches may be equally
acceptable. This is particularly true for those problems requesting a counterexample. I have not tried
AI
to be exhaustive in discussing each exercise, but rather to be suggestive.
Let me remind you that the starred exercises are not necessarily the more difficult ones. They
N
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
BO
indicates where starred exercises are used later. The following notations are used throughout this
manual:
= the set of natural numbers {1, 2, 3, 4, …}
O
= the set of rational numbers
=
ST
the set of real numbers
= “for every”
= “there exists”
ER
= “such that”
I have tried to be accurate in the preparation of this manual. Undoubtedly, however, some
mistakes will inadvertently slip by. I would appreciate having any errors in this manual or the text
brought to my attention.
Steven R. Lay
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
-2026 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS MANUAL
SOLUTIONS
ANALYSIS
MANUAL
WITH
ANALYSIS
AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOF FIFTH
TO PROOF
EDITION
FIFTH
Steven
EDITION
Lay Steven Lay
INSTRUCTOR’S
SOLUTIONS
SOLUTIONS
MANUAL ANALYSIS
MANUAL ANALYSIS
WITH AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOFTO
FIFTH
PROOF
EDITION
FIFTHSteven
EDITION
LaySteven Lay.pdf
VE
R
INSTRUCTOR’S
IF
SOLUTIONS MANUAL
IE
D
BR
A NALYSIS WITH AN
AI
N
I NTRODUCTION TO PROOF
BO
FIFTH EDITION
O
ST
ER
Steven Lay
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
VERIFIEDBRAINBOOSTER
-2026 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS MANUAL
SOLUTIONS
ANALYSIS
MANUAL
WITH
ANALYSIS
AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOF FIFTH
TO PROOF
EDITION
FIFTH
Steven
EDITION
Lay Steven Lay
,Page 2 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS
SOLUTIONS
MANUAL ANALYSIS
MANUAL ANALYSIS
WITH AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOFTO
FIFTH
PROOF
EDITION
FIFTHSteven
EDITION
LaySteven Lay.pdf
VE
R
IF
IE
D
BR
AI
N
BO
O
ST
ER
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages
in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2014, 2005, 2001 Pearson Education, Inc.
Publishing as Pearson, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-74748-8
ISBN-10: 0-321-74748-8
www.pearsonhighered.com
VERIFIEDBRAINBOOSTER
-2026 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS MANUAL
SOLUTIONS
ANALYSIS
MANUAL
WITH
ANALYSIS
AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOF FIFTH
TO PROOF
EDITION
FIFTH
Steven
EDITION
Lay Steven Lay
,Page 3 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS
SOLUTIONS
MANUAL ANALYSIS
MANUAL ANALYSIS
WITH AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOFTO
FIFTH
PROOF
EDITION
FIFTHSteven
EDITION
LaySteven Lay.pdf
VE
Instructor’s Solutions Manual
R
to accompany
IF
IE
D
BR
ANALYSIS
AI
with an Introduction
N
to Proof
BO
5th Edition
O
ST
ER
Steven R. Lay
Lee University
& RS\ULJKW3HDUVRQ( GXFDWLRQ,QF
Upper Saddle River, New Jersey 07458
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
-2026 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS MANUAL
SOLUTIONS
ANALYSIS
MANUAL
WITH
ANALYSIS
AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOF FIFTH
TO PROOF
EDITION
FIFTH
Steven
EDITION
Lay Steven Lay
, Page 4 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS
SOLUTIONS
MANUAL ANALYSIS
MANUAL ANALYSIS
WITH AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOFTO
FIFTH
PROOF
EDITION
FIFTHSteven
EDITION
LaySteven Lay.pdf
VE
2
R
IF
IE
This manual is intended to accompany the 5th edition of Analysis with an Introduction to Proof
by Steven R. Lay (Pearson, 2013). It contains solutions to nearly every exercise in the text. Those
D
exercises that have hints (or answers) in the back of the book are numbered in bold print, and the hints
BR
are included here for reference. While many of the proofs have been given in full detail, some of the
more routine proofs are only outlines. For some of the problems, other approaches may be equally
acceptable. This is particularly true for those problems requesting a counterexample. I have not tried
AI
to be exhaustive in discussing each exercise, but rather to be suggestive.
Let me remind you that the starred exercises are not necessarily the more difficult ones. They
N
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
BO
indicates where starred exercises are used later. The following notations are used throughout this
manual:
= the set of natural numbers {1, 2, 3, 4, …}
O
= the set of rational numbers
=
ST
the set of real numbers
= “for every”
= “there exists”
ER
= “such that”
I have tried to be accurate in the preparation of this manual. Undoubtedly, however, some
mistakes will inadvertently slip by. I would appreciate having any errors in this manual or the text
brought to my attention.
Steven R. Lay
Copyright © 2014 Pearson Education, Inc.
VERIFIEDBRAINBOOSTER
-2026 INSTRUCTOR’S
INSTRUCTOR’S
SOLUTIONS MANUAL
SOLUTIONS
ANALYSIS
MANUAL
WITH
ANALYSIS
AN INTRODUCTION
WITH AN INTRODUCTION
TO PROOF FIFTH
TO PROOF
EDITION
FIFTH
Steven
EDITION
Lay Steven Lay
