INSTRUCTOR’S
SOLUTIONS MANUAL % R E
A NALYSIS WITH AN % R E %RE
I NTRODUCTION TO PROOF %RE %RE
SIXTH EDITION % RE
Steven R. Lay and Richard G. Ligo
%re %re %re %re %re %re
Boston % r e Columbus % r e Indianapolis % r e New %reYork % r e San %reFrancisco % r e Upper
Saddle %reRiver %reAmsterdam
%re % r e Cape %reTown % r e Dubai % r e London % r e Madrid % r e Milan
% r e Munich % r e Paris % r e Montreal % r e Toronto
Delhi % r e Mexico %reCity % r e São %rePaulo % r e Sydney % r e Hong %reKong % r e Seoul % r e Singapore
% r e Taipei % r e Tokyo
,Product %r e Manager: % r e Evan %reSt %reCyr; %reAllison %reDeVille
Content %reAnalyst: % r e Jonathan %reKrebs
Content %reProducer: %reIshan %reChaudhary
Copyright %re© %re2024, %re2014, %re2005 %reby %rePearson %reEducation, %reInc. %reor %reits %reaffiliates. %reAll %reRights
%reReserved. %reManufactured %rein %rethe %reUnited %reStates %reof %reAmerica. %reThis %republication %reis %reprotected
%reby %recopyright, %reand %repermission %reshould %rebe %reob- %retained %refrom %rethe %republisher %reprior %reto %reany
%reprohibited %rereproduction, %restorage %rein %rea %reretrieval %resystem, %reor %retransmission %rein %reany %reform %reor
%reby %reany %remeans, %reelectronic, %remechanical, %rephotocopying, %rerecording, %reor %reotherwise. %reFor
%reinformation %reregarding %repermissions, %rerequest %reforms, %reand %rethe %reappropriate %recontacts %rewithin %rethe
%rePearson %reEducation %reGlobal %reRights %reand %rePermissions %redepartment, %replease %revisit
%rewww.pearsoned.com/permissions/.
PEARSON %reand %reMYLAB %reare %reexclusive %retrademarks %reowned %reby %rePearson %reEducation, %reInc. %reor %reits %reaffiliates %rein
%rethe
U.S. %reand/or %reother %recountries.
Unless %reotherwise %reindicated %reherein, %reany %rethird-party %retrademarks, %relogos, %reor %reicons %rethat %remay
%reappear %rein %rethis %rework %reare %rethe %reproperty %reof %retheir %rerespective %reowners, %reand %reany %rereferences
%reto %rethird-party %retrademarks, %relogos, %reicons, %reor %reother %retrade %redress %reare %refor %redemonstrative %reor
%redescriptive %repurposes %reonly. %reSuch %rereferences %reare %renot %reintended %reto %reim- %reply %reany
%responsorship, %reendorsement, %reauthorization, %reor %repromotion %reof %rePearson’s %reproducts %reby %rethe
%reowners %reof %resuch %remarks, %reor %reany %rerelationship %rebetween %rethe %reowner %reand %rePearson %reEducation,
%reInc., %reor %reits %reaffiliates, %reauthors, %relicensees, %reor %redistributors.
,Instructor’s Solutions Manual
%re %re
to %re accompany
ANALYSIS
with an %re
Introduction to
%re %re
Proof
%re
6th Edition
%re
Steven R. Lay and Richard G. Ligo
%re %re %re %re %re %re
Lee %reUniversity Gannon %reUniversity
© %reCopyright %re2024 %reby %rePearson %reEducation, %reInc. %reor %reits %reaffiliates
Upper %reSaddle %reRiver, %reNew %reJersey %re07458
Copyright %re© %re2024 %reby %rePearson %reEducation, %reInc.
%reor %reits %reaffiliates
, 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
Copyright © 2024 by Pearson Education, Inc. or its affiliates
SOLUTIONS MANUAL % R E
A NALYSIS WITH AN % R E %RE
I NTRODUCTION TO PROOF %RE %RE
SIXTH EDITION % RE
Steven R. Lay and Richard G. Ligo
%re %re %re %re %re %re
Boston % r e Columbus % r e Indianapolis % r e New %reYork % r e San %reFrancisco % r e Upper
Saddle %reRiver %reAmsterdam
%re % r e Cape %reTown % r e Dubai % r e London % r e Madrid % r e Milan
% r e Munich % r e Paris % r e Montreal % r e Toronto
Delhi % r e Mexico %reCity % r e São %rePaulo % r e Sydney % r e Hong %reKong % r e Seoul % r e Singapore
% r e Taipei % r e Tokyo
,Product %r e Manager: % r e Evan %reSt %reCyr; %reAllison %reDeVille
Content %reAnalyst: % r e Jonathan %reKrebs
Content %reProducer: %reIshan %reChaudhary
Copyright %re© %re2024, %re2014, %re2005 %reby %rePearson %reEducation, %reInc. %reor %reits %reaffiliates. %reAll %reRights
%reReserved. %reManufactured %rein %rethe %reUnited %reStates %reof %reAmerica. %reThis %republication %reis %reprotected
%reby %recopyright, %reand %repermission %reshould %rebe %reob- %retained %refrom %rethe %republisher %reprior %reto %reany
%reprohibited %rereproduction, %restorage %rein %rea %reretrieval %resystem, %reor %retransmission %rein %reany %reform %reor
%reby %reany %remeans, %reelectronic, %remechanical, %rephotocopying, %rerecording, %reor %reotherwise. %reFor
%reinformation %reregarding %repermissions, %rerequest %reforms, %reand %rethe %reappropriate %recontacts %rewithin %rethe
%rePearson %reEducation %reGlobal %reRights %reand %rePermissions %redepartment, %replease %revisit
%rewww.pearsoned.com/permissions/.
PEARSON %reand %reMYLAB %reare %reexclusive %retrademarks %reowned %reby %rePearson %reEducation, %reInc. %reor %reits %reaffiliates %rein
%rethe
U.S. %reand/or %reother %recountries.
Unless %reotherwise %reindicated %reherein, %reany %rethird-party %retrademarks, %relogos, %reor %reicons %rethat %remay
%reappear %rein %rethis %rework %reare %rethe %reproperty %reof %retheir %rerespective %reowners, %reand %reany %rereferences
%reto %rethird-party %retrademarks, %relogos, %reicons, %reor %reother %retrade %redress %reare %refor %redemonstrative %reor
%redescriptive %repurposes %reonly. %reSuch %rereferences %reare %renot %reintended %reto %reim- %reply %reany
%responsorship, %reendorsement, %reauthorization, %reor %repromotion %reof %rePearson’s %reproducts %reby %rethe
%reowners %reof %resuch %remarks, %reor %reany %rerelationship %rebetween %rethe %reowner %reand %rePearson %reEducation,
%reInc., %reor %reits %reaffiliates, %reauthors, %relicensees, %reor %redistributors.
,Instructor’s Solutions Manual
%re %re
to %re accompany
ANALYSIS
with an %re
Introduction to
%re %re
Proof
%re
6th Edition
%re
Steven R. Lay and Richard G. Ligo
%re %re %re %re %re %re
Lee %reUniversity Gannon %reUniversity
© %reCopyright %re2024 %reby %rePearson %reEducation, %reInc. %reor %reits %reaffiliates
Upper %reSaddle %reRiver, %reNew %reJersey %re07458
Copyright %re© %re2024 %reby %rePearson %reEducation, %reInc.
%reor %reits %reaffiliates
, 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
Copyright © 2024 by Pearson Education, Inc. or its affiliates
