@Aplusstuvia
INSTRUCTOR’S
SOLUTIONS MANUAL
A NALYSIS WITH AN
I NTRODUCTION TO PROOF
Sixth EDITION
Steven R. Lay and Richard G. Ligo
@
A
pl
us
stu
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
vi
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
a
@Aplusstuvia
, @Aplusstuvia
Instructor’s Solutions Manual
to accompany
ANALYSIS
with an Introduction
to Proof
6th Edition
Steven R. Lay and Richard G. Ligo
Lee University Gannon University
@
A
© Copyright 2024 by Pearson Education, Inc. or its affiliates
pl
us
Upper Saddle River, New Jersey 07458
stu
vi
a
Copyright © 2024 by Pearson Education, Inc. or its affiliates
@Aplusstuvia
, @Aplusstuvia
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.
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
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
R = the set of real numbers
∀ = “for every”
∃ = “there exists”
∋ = “such that”
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
attention.
Richard G. Ligo
@
A
pl
us
stu
vi
a
Copyright © 2024 by Pearson Education, Inc. or its affiliates
@Aplusstuvia
, @Aplusstuvia
3
Table of Starred Exercises
Note: The prefix P indicates a practice problem, the prefix E indicates an example, the prefix T refers to a
theorem or corollary, and the absence of a prefix before a number indicates an exercise.
Starred Starred
Exercise Later Use Exercise Later use
2.1.26 T3.4.11 4.3.14 4.4.5
2.2.10 2.4.26 4.4.10 8.2.15
2.3.32 2.5.3 4.4.16 8.3.10
3.1.3 E7.1.8 4.4.17 T8.3.4
3.1.4 7.1.7 5.1.16 6.2.8
3.1.6 E8.1.2 5.1.18 T6.2.12
3.1.7 4.3.10, 4.3.15, E8.1.10, T9.2.11 5.1.20 5.2.14, 5.3.17
3.1.8 P8.1.5 5.1.21 5.2.17
3.1.24 4.1.7f, E5.3.8 5.2.10 T7.2.8
3.1.27 3.3.16 5.2.11 7.2.9b
3.1.30b 3.3.13, E4.1.12, 4.3.15 5.2.13 T5.3.6, T6.1.7, 7.1.14
3.2.6a 4.1.9a, T4.2.1, 6.2.23, 7.2.16, T9.2.11 5.2.16 9.2.15
3.2.6b T6.3.8 5.3.14b T6.2.11, T6.2.13
3.2.6c T4.1.16 6.1.6 6.2.14, 6.2.19
3.2.7 T8.2.10 6.1.8 7.3.15
3.3.7 T7.2.4, 7.2.3 6.1.18b 6.4.10
3.3.14 7.1.15, T7.2.4 6.2.8 T7.2.1
3.4.15 3.5.13, T4.3.12 6.3.15d 9.3.16
3.4.21 3.5.8 7.1.13 P7.2.5
3.5.9 9.2.15 7.1.14 7.2.5
@
3.6.13 5.5.9 7.1.17 7.2.17
4.1.6b E4.2.2 7.2.9a P7.3.7
4.1.7f T4.2.7, 4.3.10, E8.1.10 7.2.11 T8.2.6
A
4.1.9a 5.2.10, 9.2.17 7.2.15 7.3.22
4.1.11 E4.3.4 7.2.20 E7.3.9
4.1.12 5.1.17 8.1.7 E8.2.11
pl
4.1.13 5.1.15 8.1.8 8.2.14
4.1.15b 4.4.11, 4.4.18, 5.3.13 8.1.13a 9.3.8
us
4.1.16 5.1.17 8.2.12 9.2.7, 9.2.8
4.2.19 E6.4.6 8.2.15 T8.3.6
4.2.20 5.1.16, T9.1.10 9.1.15a 9.2.9
stu
vi
a
Copyright © 2024 by Pearson Education, Inc. or its affiliates
@Aplusstuvia
INSTRUCTOR’S
SOLUTIONS MANUAL
A NALYSIS WITH AN
I NTRODUCTION TO PROOF
Sixth EDITION
Steven R. Lay and Richard G. Ligo
@
A
pl
us
stu
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
vi
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
a
@Aplusstuvia
, @Aplusstuvia
Instructor’s Solutions Manual
to accompany
ANALYSIS
with an Introduction
to Proof
6th Edition
Steven R. Lay and Richard G. Ligo
Lee University Gannon University
@
A
© Copyright 2024 by Pearson Education, Inc. or its affiliates
pl
us
Upper Saddle River, New Jersey 07458
stu
vi
a
Copyright © 2024 by Pearson Education, Inc. or its affiliates
@Aplusstuvia
, @Aplusstuvia
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.
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
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
R = the set of real numbers
∀ = “for every”
∃ = “there exists”
∋ = “such that”
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
attention.
Richard G. Ligo
@
A
pl
us
stu
vi
a
Copyright © 2024 by Pearson Education, Inc. or its affiliates
@Aplusstuvia
, @Aplusstuvia
3
Table of Starred Exercises
Note: The prefix P indicates a practice problem, the prefix E indicates an example, the prefix T refers to a
theorem or corollary, and the absence of a prefix before a number indicates an exercise.
Starred Starred
Exercise Later Use Exercise Later use
2.1.26 T3.4.11 4.3.14 4.4.5
2.2.10 2.4.26 4.4.10 8.2.15
2.3.32 2.5.3 4.4.16 8.3.10
3.1.3 E7.1.8 4.4.17 T8.3.4
3.1.4 7.1.7 5.1.16 6.2.8
3.1.6 E8.1.2 5.1.18 T6.2.12
3.1.7 4.3.10, 4.3.15, E8.1.10, T9.2.11 5.1.20 5.2.14, 5.3.17
3.1.8 P8.1.5 5.1.21 5.2.17
3.1.24 4.1.7f, E5.3.8 5.2.10 T7.2.8
3.1.27 3.3.16 5.2.11 7.2.9b
3.1.30b 3.3.13, E4.1.12, 4.3.15 5.2.13 T5.3.6, T6.1.7, 7.1.14
3.2.6a 4.1.9a, T4.2.1, 6.2.23, 7.2.16, T9.2.11 5.2.16 9.2.15
3.2.6b T6.3.8 5.3.14b T6.2.11, T6.2.13
3.2.6c T4.1.16 6.1.6 6.2.14, 6.2.19
3.2.7 T8.2.10 6.1.8 7.3.15
3.3.7 T7.2.4, 7.2.3 6.1.18b 6.4.10
3.3.14 7.1.15, T7.2.4 6.2.8 T7.2.1
3.4.15 3.5.13, T4.3.12 6.3.15d 9.3.16
3.4.21 3.5.8 7.1.13 P7.2.5
3.5.9 9.2.15 7.1.14 7.2.5
@
3.6.13 5.5.9 7.1.17 7.2.17
4.1.6b E4.2.2 7.2.9a P7.3.7
4.1.7f T4.2.7, 4.3.10, E8.1.10 7.2.11 T8.2.6
A
4.1.9a 5.2.10, 9.2.17 7.2.15 7.3.22
4.1.11 E4.3.4 7.2.20 E7.3.9
4.1.12 5.1.17 8.1.7 E8.2.11
pl
4.1.13 5.1.15 8.1.8 8.2.14
4.1.15b 4.4.11, 4.4.18, 5.3.13 8.1.13a 9.3.8
us
4.1.16 5.1.17 8.2.12 9.2.7, 9.2.8
4.2.19 E6.4.6 8.2.15 T8.3.6
4.2.20 5.1.16, T9.1.10 9.1.15a 9.2.9
stu
vi
a
Copyright © 2024 by Pearson Education, Inc. or its affiliates
@Aplusstuvia
