Table Of ContentIntroductory Analysis
Introductory Analysis
An Inquiry Approach
John D. Ross
Southwestern University
Kendall C. Richards
Southwestern University
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Contents
Preface ix
Prerequisites 1
P1Exploring Mathematical Statements 3
P1.1 What is a mathematical statement? . . . . . . . . . . . . . . 3
P1.2 Basic set theory . . . . . . . . . . . . . . . . . . . . . . . . . 5
P1.3 Quantifiers, both existential and universal . . . . . . . . . . . 10
P1.4 Implication: the heart of a “provable” mathematical
statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
P1.5 Negations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
P1.6 Statements related to implication . . . . . . . . . . . . . . . 17
P2 Proving Mathematical Statements 21
P2.1 Using definitions . . . . . . . . . . . . . . . . . . . . . . . . . 21
P2.2 Proving a basic statement with an existential quantifier . . . 22
P2.3 Proving a basic statement with a universal quantifier . . . . 23
P2.4 Proving an implication directly . . . . . . . . . . . . . . . . . 24
P2.5 Proof by contrapositive . . . . . . . . . . . . . . . . . . . . . 26
P2.6 Proof involving cases . . . . . . . . . . . . . . . . . . . . . . 27
P2.7 Proof by contradiction . . . . . . . . . . . . . . . . . . . . . 28
P2.8 Proof by induction . . . . . . . . . . . . . . . . . . . . . . . . 29
P2.9 Proving that one of two (or one of several) conclusions
is true . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
P3 Preliminary Content 35
P3.1 Relations and equivalence . . . . . . . . . . . . . . . . . . . . 35
P3.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
P3.3 Inequalities and epsilons . . . . . . . . . . . . . . . . . . . . 41
v
vi Contents
Main Content 43
1 Properties of R 45
1.1 Preliminary work . . . . . . . . . . . . . . . . . . . . . . . . 46
1.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.3 Follow-up work . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2 Accumulation Points and Closed Sets 59
2.1 Preliminary work . . . . . . . . . . . . . . . . . . . . . . . . 59
2.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.3 Follow-up work . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Open Sets and Open Covers 65
3.1 Preliminary work . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Follow-up work . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Sequences and Convergence 71
4.1 Preliminary work . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Follow-up work . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Subsequences and Cauchy Sequences 83
5.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Functions, Limits, and Continuity 93
6.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 98
7 Connected Sets and the Intermediate Value Theorem 101
7.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 104
Contents vii
8 Compact Sets 109
8.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 113
9 Uniform Continuity 115
9.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 115
9.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 118
10 Introduction to the Derivative 119
10.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 119
10.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 123
11 The Extreme and Mean Value Theorems 125
11.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 125
11.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 126
11.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 128
12 The Definite Integral: Part I 133
12.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 134
12.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 138
12.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 139
13 The Definite Integral: Part II 143
13.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 143
13.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 145
13.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 148
14 The Fundamental Theorem(s) of Calculus 151
14.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . 151
14.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 152
14.3 Follow-up Work . . . . . . . . . . . . . . . . . . . . . . . . . 156
15 Series 159
15.1 Preliminary work . . . . . . . . . . . . . . . . . . . . . . . . 160
15.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 162
15.3 Follow-up work . . . . . . . . . . . . . . . . . . . . . . . . . . 170
viii Contents
Extended Explorations 175
E1Function Approximation 177
E1.1 Taylor Polynomials and Taylor’s Theorem . . . . . . . . . . 177
E1.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
E1.3 Divided Differences . . . . . . . . . . . . . . . . . . . . . . . 184
E1.4 A Hybrid Approach . . . . . . . . . . . . . . . . . . . . . . . 185
E2Power Series 187
E2.1 Introduction to Power Series . . . . . . . . . . . . . . . . . . 187
E2.2 Differentiation of a Power Series . . . . . . . . . . . . . . . . 188
E2.3 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
E3Sequences and Series of Functions 199
E3.1 Pointwise Convergence . . . . . . . . . . . . . . . . . . . . . 199
E3.2 Uniform Convergence and Uniformly Cauchy
Sequences of Functions . . . . . . . . . . . . . . . . . . . . . 200
E3.3 Consequences of Uniform Convergence . . . . . . . . . . . . 202
E4Metric Spaces 207
E4.1 What is a Metric Space? Examples . . . . . . . . . . . . . . 207
E4.2 Metric Space Completeness . . . . . . . . . . . . . . . . . . . 210
E4.3 Metric Space Compactness . . . . . . . . . . . . . . . . . . . 213
E5Iterated Functions and Fixed Point Theorems 219
E5.1 Iterative Maps and Fixed Points . . . . . . . . . . . . . . . . 219
E5.2 Contraction Mappings . . . . . . . . . . . . . . . . . . . . . . 223
E5.3 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . 225
Appendix 229
A Brief Summary of Ordered Field Properties 231
Bibliography 233
Index 235
Preface
In recent years there have been mounting calls to transform the mathematics
classroom into an active and engaging space. The Common Vision project,
which synthesizes and promotes evidence-based teaching practices across the
majormathematicalassociationsintheUnitedStates,stressestheimportance
of math classes “moving toward environments that incorporate multiple ped-
agogical approaches” such as “active learning models where students engage
inactivitiessuchasreading,writing,discussion,orproblemsolving”[1].This
text aims to offer a self-contained introduction to undergraduate elementary
real analysis in a manner that promotes active learning.
Undergraduate real analysis exposes students to most of the large ideas
present in Calculus, but at a much higher level of rigor than in a traditional
Calculus classes. Perhaps more importantly, students are also introduced to
a host of logical reasoning skills and proof-writing techniques. Gaining this
bodyofknowledgeservesasanimportantmilestoneinthemathematicsmajor
and can act as a launchpad to future math classes. However, to achieve its
full potential, an undergraduate analysis class must (a) cover a broad range
of material, and (b) offer suitable opportunities for the student to practice
and develop their skills of proof-writing and proof-analyzing. These two goals
oftenexistintensionwitheachother,asthereisonlyafinite(indeed,ashort)
amount of time to accomplish both.
Our intent in writing this text has been to find a “sweet spot” between
pure Inquiry-Based Learning and more traditional approaches. The primary
aim of this text is to provide a self-contained introduction to elementary real
analysisusinganinquiry-orientedapproachwithscaffolding.Thepresentation
is intended to be “inquiry-oriented” in the sense that, as each major topic
is discussed, details of the proofs are left to the student. However, we do
include scaffolding for most of the theorems. Our scaffolding takes the form
of brief guiding prompts marked as Key Steps in a Proof. Students are
asked to follow these guidelines and work toward a prompted, but largely
self-generated, proof of each theorem. The symbol (cid:32) is included when we
want to particularly emphasize that there is work that must be done by the
student in order to complete the specific stage of the argument. We close the
scaffolding discussion with the symbol (cid:13) and restrict the use of the symbol
(cid:3)
to only those cases where a complete proof is provided.
Although the mathematics covered here is not new (almost all of it has
been known for hundreds of years), it is hoped that the text is framed in
ix
