Instructor’s Solution Manual
l l
INTRODUCTION l
TO REAL ANALYSIS
l l
William F. Trench l l
Professorl Emeritus
TrinitylUniversitylSan
lAntonio,lTexas,lUSA
©Copyrightl2009lWilliamlF.lTrench,l alll rightslreservedlUpdatedl
Mayl2012
Nolpartloflthisldocumentlmaylbelcirculatedlorlpostedlonlanylwebsitelwithoutlt
helauthor’slpermission.l UnderlUSlcopyrightllaw,
“Uploadinglorldownloadinglworkslprotectedlbylcopyrightlwithoutlthelau-
lthorityloflthelcopyrightlownerlislanlinfringementloflthelcopyrightlowner’slex
clusivelrightsloflreproductionland/orldistribution.lAnyonelfoundltolhavelinf
ringedlalcopyrightedlworklmaylbelliablelforlstatutoryldamageslupltol$30,000l
forleachlworklinfringedland,liflwillfullinfringementlislprovenlbylthelcopyrigh
tlowner,lthatlamountlmaylbelincreasedlupltol$150,000lforleachlworklinfringe
d.lInladdition,lanlinfringerloflalworklmaylalsolbelli-
lablelforlthelattorney’slfeeslincurredlbylthelcopyrightlownerltolenforcelhislorlh
erl rights.”
, Contents
Chapterl 1 Thel Reall Numbers 1
1.1 Thel Reall Numberl System 1
1.2 Mathematicall Induction 4
1.3 Thel Reall Line 13
Chapterl 2 Differentiall Calculusl ofl Functionsl ofl Onel Variablel 17
2.1 Functionsl andl Limits 17
2.2 Continuity 24
2.3 Differentiablel Functionsl ofl Onel Variable 30
2.4 L’Hospital’sl Rule 36
2.5 Taylor’slTheorem 43
Chapterl 3 IntegrallCalculusloflFunctionsloflOnelVariable 53
3.1 Definitionl ofl thel Integral 53
3.2 Existencel ofl thel Integral 56
3.3 Propertiesl ofl thel Integral 61
3.4 Improperl Integrals 66
3.5 AlMorelAdvancedlLooklatlthelExistence
ofl thel Properl Riemannl Integral 77
Chapterl 4 Infinitel Sequencesl andl Series 79
4.1 Sequencesl ofl Reall Numbers 79
4.2 EarlierlTopicsl Revisitedl Withl Sequences 87
4.3 Infinitel Seriesl ofl Constants 89
4.4 Sequencesl andl Seriesl ofl Functions 100
4.5 Powerl Series 107
,Chapterl 5 Real-Valuedl Functionsl ofl Severall Variables 116
5.1 Structurel ofl Rn 116
5.2 Continuousl Real-Valuedl Functionl ofl nl Variables 121
5.3 PartiallDerivativesl andl thel Differential 123
5.4 Thel Chainl Rulel andl Taylor’slTheorem 130
Chapterl 6 Vector-ValuedlFunctionsloflSeverallVariables 141
6.1 Linearl Transformationsl andl Matrices 141
6.2 Continuityl andl Differentiabilityl ofl Transformations 146
6.3 Thel Inversel Functionl Theorem 152
6.4 Thel Implicitl Functionl Theorem 160
Chapterl 7 Integralsl ofl Functionsl ofl SeverallVariables 170
7.1ll Definitionlandl Existencelofl thel MultiplelIntegral 170
7.2l Iteratedl Integralslandl MultiplelIntegrals 187
7.3l Changel ofl Variablesl inl Multiplel Integrals 207
Chapterl 8 Metricl Spaces 217
8.1 Introductionl tol Metricl Spaces 217
8.2 Compactl Setsl inl al Metricl Space 224
8.3 Continuousl Functionsl onl Metricl Spaces 226
, Sectionl1.1lThelReall NumberlSystem 1
CHAPTER 1 l
THE REAL NUMBERS l l
1.1 THElREALl NUMBERlSYSTEM
1:1:1.l Notel thatl jal —lbjl Dl max.a;lb/l —lmin.a;lb/.
(a) al Clbl Cljal —lbjlDlal Clbl Clmax.a;lb/l —lmin.a;lb/l Dl2lmax.a;lb/.
(b) alClbl —ljal—lbjlDlal Clbl —lmax.a;lb/lClmin.a;lb/l Dl2lmin.a;lb/.
(c) Letl˛ DlllaClllllbC
llll2c Cjla—lll bl jCl ˇC
alllll b—lll 2c Cjla— j.lˇFroml(a),l˛ D 2lŒmax.a;lb/ ClllcClj
lll bl lll max.a;lb/ —cl llj]l D df
ˇ.lllFroml (a)l withl al andl bl replacedl byl max.a;lb/l andl c,l ˇ D 4lmaxl.max.a;lb/;lc/ D
4lmax.a;lb;lc/.
(d) Letl˛ DlllaClllllbCllll2c lll b l j—ˇlaC
—jal — llllb l—
lll 2c —jal —
lll bllllj.lˇFroml(b),l˛ D 2lŒmin.a;lb/ Cl c—lj min.a;lb/ —cl llj]l D df
ˇ.lll Froml (a)l withl al andl bl replacedl byl min.a;lb/l andl c,l ˇ D 4lminl.min.a;lb/;lc/ D
4lmin.a;lb;lc/.
1:1:2.lFirstlverifylaxiomslA-E:
AxiomlA.lSeelEqns.l(1.1.1)landl(1.1.2).
AxiomlB.l Ifl a D 0l thenl .a C b/ C c D b C cl andl a .bC c/ C b c,lDsol .aC b/lllllllllllllllllllllllC
c C
a
D .b C c/.lC Similarl argumentsl applyl ifl b 0l orl c D D lremaininglcasel isla b c
0.lThe
D 1.lDSinceDl .1 1/ C 1 C 0 D 1 C 1land D l1 C.1 C 1/ D 1 C 0 D
1,ladditionlislassociative.l Since
l 0;ll unlesslal Dlbl Dlcl Dl1;
.ab/c a.bc/
D D 1;llliflal Dlbl Dlcl Dl1;
multiplicationlislassociative.l
AxiomlC.lSince
0; iflal Dl0;
a.bl c/ ab ac
C D C bl Clc;llll iflal Dl1;
theldistributivellawlholds. D
AxiomlD.lEqns.l(1.1.1)landl(1.1.2)limplylthatl0landl1lhavelthelrequiredlproperties.
l l
INTRODUCTION l
TO REAL ANALYSIS
l l
William F. Trench l l
Professorl Emeritus
TrinitylUniversitylSan
lAntonio,lTexas,lUSA
©Copyrightl2009lWilliamlF.lTrench,l alll rightslreservedlUpdatedl
Mayl2012
Nolpartloflthisldocumentlmaylbelcirculatedlorlpostedlonlanylwebsitelwithoutlt
helauthor’slpermission.l UnderlUSlcopyrightllaw,
“Uploadinglorldownloadinglworkslprotectedlbylcopyrightlwithoutlthelau-
lthorityloflthelcopyrightlownerlislanlinfringementloflthelcopyrightlowner’slex
clusivelrightsloflreproductionland/orldistribution.lAnyonelfoundltolhavelinf
ringedlalcopyrightedlworklmaylbelliablelforlstatutoryldamageslupltol$30,000l
forleachlworklinfringedland,liflwillfullinfringementlislprovenlbylthelcopyrigh
tlowner,lthatlamountlmaylbelincreasedlupltol$150,000lforleachlworklinfringe
d.lInladdition,lanlinfringerloflalworklmaylalsolbelli-
lablelforlthelattorney’slfeeslincurredlbylthelcopyrightlownerltolenforcelhislorlh
erl rights.”
, Contents
Chapterl 1 Thel Reall Numbers 1
1.1 Thel Reall Numberl System 1
1.2 Mathematicall Induction 4
1.3 Thel Reall Line 13
Chapterl 2 Differentiall Calculusl ofl Functionsl ofl Onel Variablel 17
2.1 Functionsl andl Limits 17
2.2 Continuity 24
2.3 Differentiablel Functionsl ofl Onel Variable 30
2.4 L’Hospital’sl Rule 36
2.5 Taylor’slTheorem 43
Chapterl 3 IntegrallCalculusloflFunctionsloflOnelVariable 53
3.1 Definitionl ofl thel Integral 53
3.2 Existencel ofl thel Integral 56
3.3 Propertiesl ofl thel Integral 61
3.4 Improperl Integrals 66
3.5 AlMorelAdvancedlLooklatlthelExistence
ofl thel Properl Riemannl Integral 77
Chapterl 4 Infinitel Sequencesl andl Series 79
4.1 Sequencesl ofl Reall Numbers 79
4.2 EarlierlTopicsl Revisitedl Withl Sequences 87
4.3 Infinitel Seriesl ofl Constants 89
4.4 Sequencesl andl Seriesl ofl Functions 100
4.5 Powerl Series 107
,Chapterl 5 Real-Valuedl Functionsl ofl Severall Variables 116
5.1 Structurel ofl Rn 116
5.2 Continuousl Real-Valuedl Functionl ofl nl Variables 121
5.3 PartiallDerivativesl andl thel Differential 123
5.4 Thel Chainl Rulel andl Taylor’slTheorem 130
Chapterl 6 Vector-ValuedlFunctionsloflSeverallVariables 141
6.1 Linearl Transformationsl andl Matrices 141
6.2 Continuityl andl Differentiabilityl ofl Transformations 146
6.3 Thel Inversel Functionl Theorem 152
6.4 Thel Implicitl Functionl Theorem 160
Chapterl 7 Integralsl ofl Functionsl ofl SeverallVariables 170
7.1ll Definitionlandl Existencelofl thel MultiplelIntegral 170
7.2l Iteratedl Integralslandl MultiplelIntegrals 187
7.3l Changel ofl Variablesl inl Multiplel Integrals 207
Chapterl 8 Metricl Spaces 217
8.1 Introductionl tol Metricl Spaces 217
8.2 Compactl Setsl inl al Metricl Space 224
8.3 Continuousl Functionsl onl Metricl Spaces 226
, Sectionl1.1lThelReall NumberlSystem 1
CHAPTER 1 l
THE REAL NUMBERS l l
1.1 THElREALl NUMBERlSYSTEM
1:1:1.l Notel thatl jal —lbjl Dl max.a;lb/l —lmin.a;lb/.
(a) al Clbl Cljal —lbjlDlal Clbl Clmax.a;lb/l —lmin.a;lb/l Dl2lmax.a;lb/.
(b) alClbl —ljal—lbjlDlal Clbl —lmax.a;lb/lClmin.a;lb/l Dl2lmin.a;lb/.
(c) Letl˛ DlllaClllllbC
llll2c Cjla—lll bl jCl ˇC
alllll b—lll 2c Cjla— j.lˇFroml(a),l˛ D 2lŒmax.a;lb/ ClllcClj
lll bl lll max.a;lb/ —cl llj]l D df
ˇ.lllFroml (a)l withl al andl bl replacedl byl max.a;lb/l andl c,l ˇ D 4lmaxl.max.a;lb/;lc/ D
4lmax.a;lb;lc/.
(d) Letl˛ DlllaClllllbCllll2c lll b l j—ˇlaC
—jal — llllb l—
lll 2c —jal —
lll bllllj.lˇFroml(b),l˛ D 2lŒmin.a;lb/ Cl c—lj min.a;lb/ —cl llj]l D df
ˇ.lll Froml (a)l withl al andl bl replacedl byl min.a;lb/l andl c,l ˇ D 4lminl.min.a;lb/;lc/ D
4lmin.a;lb;lc/.
1:1:2.lFirstlverifylaxiomslA-E:
AxiomlA.lSeelEqns.l(1.1.1)landl(1.1.2).
AxiomlB.l Ifl a D 0l thenl .a C b/ C c D b C cl andl a .bC c/ C b c,lDsol .aC b/lllllllllllllllllllllllC
c C
a
D .b C c/.lC Similarl argumentsl applyl ifl b 0l orl c D D lremaininglcasel isla b c
0.lThe
D 1.lDSinceDl .1 1/ C 1 C 0 D 1 C 1land D l1 C.1 C 1/ D 1 C 0 D
1,ladditionlislassociative.l Since
l 0;ll unlesslal Dlbl Dlcl Dl1;
.ab/c a.bc/
D D 1;llliflal Dlbl Dlcl Dl1;
multiplicationlislassociative.l
AxiomlC.lSince
0; iflal Dl0;
a.bl c/ ab ac
C D C bl Clc;llll iflal Dl1;
theldistributivellawlholds. D
AxiomlD.lEqns.l(1.1.1)landl(1.1.2)limplylthatl0landl1lhavelthelrequiredlproperties.
