INSTRUCTOR’S
SOLUTIONS MANUAL
A NALYSIS WITH AN
M
I NTRODUCTION TO PROOF
ED
Sixth EDITION
C
O
N
N
O
Steven R. Lay and Richard G. Ligo
IS
SE
U
R
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
, This work is protected by United States copyright laws and is provided solely
for the use of instructors in teaching their courses and assessing student
learning. Dissemination or sale of any part of this work (including on the
M
World Wide Web) will destroy the integrity of the work and is not permit-
ted. The work and materials from it should never be made available to
students except by instructors using the accompanying text in their
ED
classes. All recipients of this work are expected to abide by these
restrictions and to honor the intended pedagogical purposes and the needs of
other instructors who rely on these materials.
C
O
Product Manager: Evan St Cyr; Allison DeVille
N
Content Analyst: Jonathan Krebs
Content Producer: Ishan Chaudhary
N
O
Copyright © 2024, 2014, 2005 by Pearson Education, Inc. or its affiliates. All Rights Reserved. Manufactured
in the United States of America. This publication is protected by copyright, and permission should be ob-
tained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission
IS
in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information
regarding permissions, request forms, and the appropriate contacts within the Pearson Education Global
Rights and Permissions department, please visit www.pearsoned.com/permissions/.
SE
PEARSON and MYLAB are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the
U.S. and/or other countries.
U
Unless otherwise indicated herein, any third-party trademarks, logos, or icons that may appear in this work
are the property of their respective owners, and any references to third-party trademarks, logos, icons, or
R
other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to im-
ply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such
marks, or any relationship between the owner and Pearson Education, Inc., or its affiliates, authors, licensees,
or distributors.
@Pearson
, Instructor’s Solutions Manual
to accompany
ANALYSIS
M
with an Introduction
to Proof
ED
6th Edition
C
O
N
Steven R. Lay and Richard G. Ligo
N
Lee University Gannon University
O
IS
SE
© Copyright 2024 by Pearson Education, Inc. or its affiliates
U
R
Upper Saddle River, New Jersey 07458
Copyright © 2024 by Pearson Education, Inc. or its affiliates
, 2
This manual is intended to accompany the 6th edition of Analysis with an Introduction to Proof
by Steven R. Lay and Richard G. Ligo (Pearson, 2023). It contains solutions to nearly every exercise
in the text. Those exercises that have hints (or answers) in the back of the book are numbered in bold
print, and the hints 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.
We have not tried to be exhaustive in discussing each exercise, but rather to be suggestive.
M
Let us remind you that the starred exercises are not necessarily the more difficult ones. They
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
ED
indicates where starred exercises are used later. The following notations are used throughout this
manual:
ℕ = the set of natural numbers {1, 2, 3, 4, …}
ℚ = the set of rational numbers
C
R = the set of real numbers
∀ = “for every”
O
∃ = “there exists”
∋ = “such that”
N
We have tried to be accurate in the preparation of this manual, but unfortunately, some mistake will
inadvertently slip by. I would appreciate having any errors in this manual or the text brought to my
N
attention.
O
Richard G. Ligo
IS
SE
U
R
Copyright © 2024 by Pearson Education, Inc. or its affiliates
SOLUTIONS MANUAL
A NALYSIS WITH AN
M
I NTRODUCTION TO PROOF
ED
Sixth EDITION
C
O
N
N
O
Steven R. Lay and Richard G. Ligo
IS
SE
U
R
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
, This work is protected by United States copyright laws and is provided solely
for the use of instructors in teaching their courses and assessing student
learning. Dissemination or sale of any part of this work (including on the
M
World Wide Web) will destroy the integrity of the work and is not permit-
ted. The work and materials from it should never be made available to
students except by instructors using the accompanying text in their
ED
classes. All recipients of this work are expected to abide by these
restrictions and to honor the intended pedagogical purposes and the needs of
other instructors who rely on these materials.
C
O
Product Manager: Evan St Cyr; Allison DeVille
N
Content Analyst: Jonathan Krebs
Content Producer: Ishan Chaudhary
N
O
Copyright © 2024, 2014, 2005 by Pearson Education, Inc. or its affiliates. All Rights Reserved. Manufactured
in the United States of America. This publication is protected by copyright, and permission should be ob-
tained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission
IS
in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information
regarding permissions, request forms, and the appropriate contacts within the Pearson Education Global
Rights and Permissions department, please visit www.pearsoned.com/permissions/.
SE
PEARSON and MYLAB are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the
U.S. and/or other countries.
U
Unless otherwise indicated herein, any third-party trademarks, logos, or icons that may appear in this work
are the property of their respective owners, and any references to third-party trademarks, logos, icons, or
R
other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to im-
ply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such
marks, or any relationship between the owner and Pearson Education, Inc., or its affiliates, authors, licensees,
or distributors.
@Pearson
, Instructor’s Solutions Manual
to accompany
ANALYSIS
M
with an Introduction
to Proof
ED
6th Edition
C
O
N
Steven R. Lay and Richard G. Ligo
N
Lee University Gannon University
O
IS
SE
© Copyright 2024 by Pearson Education, Inc. or its affiliates
U
R
Upper Saddle River, New Jersey 07458
Copyright © 2024 by Pearson Education, Inc. or its affiliates
, 2
This manual is intended to accompany the 6th edition of Analysis with an Introduction to Proof
by Steven R. Lay and Richard G. Ligo (Pearson, 2023). It contains solutions to nearly every exercise
in the text. Those exercises that have hints (or answers) in the back of the book are numbered in bold
print, and the hints 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.
We have not tried to be exhaustive in discussing each exercise, but rather to be suggestive.
M
Let us remind you that the starred exercises are not necessarily the more difficult ones. They
are the exercises that are used in some way in subsequent sections. There is a table on page 3 that
ED
indicates where starred exercises are used later. The following notations are used throughout this
manual:
ℕ = the set of natural numbers {1, 2, 3, 4, …}
ℚ = the set of rational numbers
C
R = the set of real numbers
∀ = “for every”
O
∃ = “there exists”
∋ = “such that”
N
We have tried to be accurate in the preparation of this manual, but unfortunately, some mistake will
inadvertently slip by. I would appreciate having any errors in this manual or the text brought to my
N
attention.
O
Richard G. Ligo
IS
SE
U
R
Copyright © 2024 by Pearson Education, Inc. or its affiliates
