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INSTRUCTOR’S
SOLUTIONS MANUAL
A NALYSIS WITH AN
I NTRODUCTION TO PROOF
FIFTH EDITION
elitedocs
Steven Lay
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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
elitedocs elitedocs
, elitedocs
Instructor’s Solutions Manual
to accompany
ANALYSIS
with an Introduction
to Proof
5th Edition
elitedocs
Steven R. Lay
Lee University
&RS\ULJKW3HDUVRQ(GXFDWLRQ,QF
Upper Saddle River, New Jersey 07458
elitedocs
Copyright © 2014 Pearson Education, Inc.
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, elitedocs
2
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
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. I have not tried
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
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:
= the set of natural numbers {1, 2, 3, 4, …}
= the set of rational numbers
= the set of real numbers
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= “for every”
= “there exists”
= “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
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Copyright © 2014 Pearson Education, Inc.
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, elitedocs
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 8.2.14
2.3.32 2.5.3 4.4.16 8.3.9
3.1.3 E7.1.7 4.4.17 T8.3.3
3.1.4 7.1.7 5.1.14 6.2.8
3.1.6 E8.1.1 5.1.16 T6.2.9
3.1.7 4.3.10, 4.3.15, E8.1.7, T9.2.9 5.1.18 5.2.14, 5.3.15
3.1.8 P8.1.3 5.1.19 5.2.17
3.1.24 4.1.7f, E5.3.7 5.2.10 T7.2.8
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3.1.27 3.3.14 5.2.11 7.2.9b
3.1.30b 3.3.11, E4.1.11, 4.3.14 5.2.13 T5.3.5, T6.1.7, 7.1.13
3.2.6a 4.1.9a, T4.2.1, 6.2.23, 7.2.16, T9.2.9 5.2.16 9.2.15
3.2.6b T6.3.8 5.3.13b T6.2.8, T6.2.10
3.2.6c T4.1.14 6.1.6 6. 2.14, 6.2.19
3.2.7 T8.2.5 6.1.8 7.3.13
3.3.7 T7.2.4, 7.2.3 6.1.17b 6.4.9
3.3.12 7.1.14, T7.2.4 6.2.8 T7.2.1
3.4.15 3.5.12, T4.3.12 6.3.13d 9.3.16
3.4.21 3.5.7 7.1.12 P7.2.5
3.5.8 9.2.15 7.1.13 7.2.5
3.6.12 5.5.9 7.1.16 7.2.17
4.1.6b E4.2.2 7.2.9a P7.3.7
4.1.7f T4.2.7, 4.3.10, E8.1.7 7.2.11 T8.2.13
4.1.9a 5.2.10, 9.2.17 7.2.15 7.3.20
4.1.11 E4.3.4 7.2.20 E7.3.9
4.1.12 5.1.15 8.1.7 E8.2.6
4.1.13 5.1.13 8.1.8 8.2.13
4.1.15b 4.4.11, 4.4.18, 5.3.12 8.1.13a 9.3.8
4.1.16 5.1.15 8.2.12 9.2.7, 9.2.8
4.2.17 E6.4.3 8.2.14 T8.3.4
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4.2.18 5.1.14, T9.1.10 9.1.15a 9.2.9
Copyright © 2014 Pearson Education, Inc.
elitedocs elitedocs
INSTRUCTOR’S
SOLUTIONS MANUAL
A NALYSIS WITH AN
I NTRODUCTION TO PROOF
FIFTH EDITION
elitedocs
Steven Lay
elitedocs
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
elitedocs elitedocs
, elitedocs
Instructor’s Solutions Manual
to accompany
ANALYSIS
with an Introduction
to Proof
5th Edition
elitedocs
Steven R. Lay
Lee University
&RS\ULJKW3HDUVRQ(GXFDWLRQ,QF
Upper Saddle River, New Jersey 07458
elitedocs
Copyright © 2014 Pearson Education, Inc.
elitedocs elitedocs
, elitedocs
2
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
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. I have not tried
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
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:
= the set of natural numbers {1, 2, 3, 4, …}
= the set of rational numbers
= the set of real numbers
elitedocs
= “for every”
= “there exists”
= “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
elitedocs
Copyright © 2014 Pearson Education, Inc.
elitedocs elitedocs
, elitedocs
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 8.2.14
2.3.32 2.5.3 4.4.16 8.3.9
3.1.3 E7.1.7 4.4.17 T8.3.3
3.1.4 7.1.7 5.1.14 6.2.8
3.1.6 E8.1.1 5.1.16 T6.2.9
3.1.7 4.3.10, 4.3.15, E8.1.7, T9.2.9 5.1.18 5.2.14, 5.3.15
3.1.8 P8.1.3 5.1.19 5.2.17
3.1.24 4.1.7f, E5.3.7 5.2.10 T7.2.8
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3.1.27 3.3.14 5.2.11 7.2.9b
3.1.30b 3.3.11, E4.1.11, 4.3.14 5.2.13 T5.3.5, T6.1.7, 7.1.13
3.2.6a 4.1.9a, T4.2.1, 6.2.23, 7.2.16, T9.2.9 5.2.16 9.2.15
3.2.6b T6.3.8 5.3.13b T6.2.8, T6.2.10
3.2.6c T4.1.14 6.1.6 6. 2.14, 6.2.19
3.2.7 T8.2.5 6.1.8 7.3.13
3.3.7 T7.2.4, 7.2.3 6.1.17b 6.4.9
3.3.12 7.1.14, T7.2.4 6.2.8 T7.2.1
3.4.15 3.5.12, T4.3.12 6.3.13d 9.3.16
3.4.21 3.5.7 7.1.12 P7.2.5
3.5.8 9.2.15 7.1.13 7.2.5
3.6.12 5.5.9 7.1.16 7.2.17
4.1.6b E4.2.2 7.2.9a P7.3.7
4.1.7f T4.2.7, 4.3.10, E8.1.7 7.2.11 T8.2.13
4.1.9a 5.2.10, 9.2.17 7.2.15 7.3.20
4.1.11 E4.3.4 7.2.20 E7.3.9
4.1.12 5.1.15 8.1.7 E8.2.6
4.1.13 5.1.13 8.1.8 8.2.13
4.1.15b 4.4.11, 4.4.18, 5.3.12 8.1.13a 9.3.8
4.1.16 5.1.15 8.2.12 9.2.7, 9.2.8
4.2.17 E6.4.3 8.2.14 T8.3.4
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4.2.18 5.1.14, T9.1.10 9.1.15a 9.2.9
Copyright © 2014 Pearson Education, Inc.
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