Instructor’s Solution Manual
INTRODUCTION
TO REAL ANALYSIS
(Comprehensive Guide)
, Contents
Chapter 1 The Real Numbers 1
1.1 The Real Number System 1
1.2 Mathematical Induction 4
1.3 The Real Line 13
Chapter 2 Differential Calculus of Functions of One Variable 17
2.1 Functions and Limits 17
2.2 Continuity 24
2.3 Differentiable Functions of One Variable 30
2.4 L’Hospital’s Rule 36
2.5 Taylor’s Theorem 43
Chapter 3 Integral Calculus of Functions of One Variable 53
3.1 Definition of the Integral 53
3.2 Existence of the Integral 56
3.3 Properties of the Integral 61
3.4 Improper Integrals 66
3.5 A More Advanced Look at the Existence
of the Proper Riemann Integral 77
Chapter 4 Infinite Sequences and Series 79
4.1 Sequences of Real Numbers 79
4.2 Earlier Topics Revisited With Sequences 87
4.3 Infinite Series of Constants 89
4.4 Sequences and Series of Functions 100
4.5 Power Series 107
,Chapter 5 Real-Valued Functions of Several Variables 116
5.1 Structure of Rn 116
5.2 Continuous Real-Valued Function of n Variables 121
5.3 Partial Derivatives and the Differential 123
5.4 The Chain Rule and Taylor’s Theorem 130
Chapter 6 Vector-Valued Functions of Several Variables 141
6.1 Linear Transformations and Matrices 141
6.2 Continuity and Differentiability of Transformations 146
6.3 The Inverse Function Theorem 152
6.4 The Implicit Function Theorem 160
Chapter 7 Integrals of Functions of Several Variables 170
7.1 Definition and Existence of the Multiple Integral 170
7.2 Iterated Integrals and Multiple Integrals 187
7.3 Change of Variables in Multiple Integrals 207
Chapter 8 Metric Spaces 217
8.1 Introduction to Metric Spaces 217
8.2 Compact Sets in a Metric Space 224
8.3 Continuous Functions on Metric Spaces 226
, Section 1.1 The Real Number System 1
CHAPTER 1
THE REAL NUMBERS
1.1 THE REAL NUMBER SYSTEM
1:1:1. Note that ja — bj D max.a; b/ — min.a; b/.
(a) a C b C ja — bj D a C b C max.a; b/ — min.a; b/ D 2 max.a; b/.
(b) a C b — ja — bj D a C b — max.a; b/ C min.a; b/ D 2 min.a; b/.
(c) Let ˛ D a C b C 2c Cj a—b jC ˇaC b—2c Cj a—b jˇ. From (a), ˛ D 2 Œmax.a; b/ C c C j max.a; b/ —c j] D df
ˇ. From (a) with a and b replaced by max.a; b/ and c, ˇ D 4 max .max.a; b/; c/ D
4 max.a; b; c/.
(d) Let ˛ D a C b C 2c —ja —b j—ˇa Cb—2c —ja —b jˇ. From (b), ˛ D 2 Œmin.a; b/ C c — j min.a; b/ —c j] D df
ˇ. From (a) with a and b replaced by min.a; b/ and c, ˇ D 4 min .min.a; b/; c/ D
4 min.a; b; c/.
1:1:2. First verify axioms A-E:
Axiom A. See Eqns. (1.1.1) and (1.1.2).
Axiom B. If a D 0 then .a C b/ C c D b C c and a.bC c/ C b c, D
so .aC b/ C C
c D a C .b C c/. Similar arguments apply if b D 0 or c D 0. The remaining case is
a D b D c D 1. Since .1 C 1/ C 1 D 0 C 1 D 1 and 1 C .1 C 1/ D 1 C 0 D 1, addition
is associative. Since
0; unless a D b D c D 1;
.ab/c D a.bc/ D
1; if a D b D c D 1;
multiplication is associative.
Axiom C. Since
0; if a D 0;
a.b C c/ ab
D ac C
b C c; if a D 1;
D
the distributive law holds.
Axiom D. Eqns. (1.1.1) and (1.1.2) imply that 0 and 1 have the required properties.
INTRODUCTION
TO REAL ANALYSIS
(Comprehensive Guide)
, Contents
Chapter 1 The Real Numbers 1
1.1 The Real Number System 1
1.2 Mathematical Induction 4
1.3 The Real Line 13
Chapter 2 Differential Calculus of Functions of One Variable 17
2.1 Functions and Limits 17
2.2 Continuity 24
2.3 Differentiable Functions of One Variable 30
2.4 L’Hospital’s Rule 36
2.5 Taylor’s Theorem 43
Chapter 3 Integral Calculus of Functions of One Variable 53
3.1 Definition of the Integral 53
3.2 Existence of the Integral 56
3.3 Properties of the Integral 61
3.4 Improper Integrals 66
3.5 A More Advanced Look at the Existence
of the Proper Riemann Integral 77
Chapter 4 Infinite Sequences and Series 79
4.1 Sequences of Real Numbers 79
4.2 Earlier Topics Revisited With Sequences 87
4.3 Infinite Series of Constants 89
4.4 Sequences and Series of Functions 100
4.5 Power Series 107
,Chapter 5 Real-Valued Functions of Several Variables 116
5.1 Structure of Rn 116
5.2 Continuous Real-Valued Function of n Variables 121
5.3 Partial Derivatives and the Differential 123
5.4 The Chain Rule and Taylor’s Theorem 130
Chapter 6 Vector-Valued Functions of Several Variables 141
6.1 Linear Transformations and Matrices 141
6.2 Continuity and Differentiability of Transformations 146
6.3 The Inverse Function Theorem 152
6.4 The Implicit Function Theorem 160
Chapter 7 Integrals of Functions of Several Variables 170
7.1 Definition and Existence of the Multiple Integral 170
7.2 Iterated Integrals and Multiple Integrals 187
7.3 Change of Variables in Multiple Integrals 207
Chapter 8 Metric Spaces 217
8.1 Introduction to Metric Spaces 217
8.2 Compact Sets in a Metric Space 224
8.3 Continuous Functions on Metric Spaces 226
, Section 1.1 The Real Number System 1
CHAPTER 1
THE REAL NUMBERS
1.1 THE REAL NUMBER SYSTEM
1:1:1. Note that ja — bj D max.a; b/ — min.a; b/.
(a) a C b C ja — bj D a C b C max.a; b/ — min.a; b/ D 2 max.a; b/.
(b) a C b — ja — bj D a C b — max.a; b/ C min.a; b/ D 2 min.a; b/.
(c) Let ˛ D a C b C 2c Cj a—b jC ˇaC b—2c Cj a—b jˇ. From (a), ˛ D 2 Œmax.a; b/ C c C j max.a; b/ —c j] D df
ˇ. From (a) with a and b replaced by max.a; b/ and c, ˇ D 4 max .max.a; b/; c/ D
4 max.a; b; c/.
(d) Let ˛ D a C b C 2c —ja —b j—ˇa Cb—2c —ja —b jˇ. From (b), ˛ D 2 Œmin.a; b/ C c — j min.a; b/ —c j] D df
ˇ. From (a) with a and b replaced by min.a; b/ and c, ˇ D 4 min .min.a; b/; c/ D
4 min.a; b; c/.
1:1:2. First verify axioms A-E:
Axiom A. See Eqns. (1.1.1) and (1.1.2).
Axiom B. If a D 0 then .a C b/ C c D b C c and a.bC c/ C b c, D
so .aC b/ C C
c D a C .b C c/. Similar arguments apply if b D 0 or c D 0. The remaining case is
a D b D c D 1. Since .1 C 1/ C 1 D 0 C 1 D 1 and 1 C .1 C 1/ D 1 C 0 D 1, addition
is associative. Since
0; unless a D b D c D 1;
.ab/c D a.bc/ D
1; if a D b D c D 1;
multiplication is associative.
Axiom C. Since
0; if a D 0;
a.b C c/ ab
D ac C
b C c; if a D 1;
D
the distributive law holds.
Axiom D. Eqns. (1.1.1) and (1.1.2) imply that 0 and 1 have the required properties.
