Introduction to Analysis [Lecture notes]

359 Pages·2017·2.012 MB·English

by  Irena Swanson

Introdu
tion to Analysis
Irena Swanson
Reed College
Spring 2016

Table of contents
Preface 7
The briefest overview, motivation, notation 9
Chapter 1: How we will do mathematics 13
Section 1.1: Statements and proof methods 13
Section 1.2: Statements with quantifiers 24
Section 1.3: More proof methods, and negation 27
Section 1.4: Summation 34
Section 1.5: Proofs by (mathematical) induction 36
Section 1.6: Pascal’s triangle 45
Chapter 2: Concepts with which we will do mathematics 49
Section 2.1: Sets 49
Section 2.2: Cartesian product 57
Section 2.3: Relations, equivalence relations 59
Section 2.4: Functions 65
Section 2.5: Binary operations 74
Section 2.6: Fields 80
Section 2.7: Order on sets, ordered fields 83
Section 2.8: Increasing and decreasing functions 90
Section 2.9: Absolute values 92
Chapter 3: Construction of the basic number systems 96
Section 3.1: Inductive sets and a construction of natural numbers 96
Section 3.2: Arithmetic on N 101
0
Section 3.3: Order on N 104
0
Section 3.4: Cancellation in N 106
0
Section 3.5: Construction of Z, arithmetic, and order on Z 108
Section 3.6: Construction of the ordered field Q of rational numbers 114
Section 3.7: Construction of the field R of real numbers 118
Section 3.8: Order on R, the least upper/greatest lower bound theorem 126
4
Section 3.9: Complex numbers 129
Section 3.10: Functions related to complex numbers 132
Section 3.11: Absolute value in C 133
Section 3.12: Polar coordinates 137
Section 3.13: Topology on the fields of real and complex numbers 142
Section 3.14: Closed and bounded sets, and open balls 148
Chapter 4: Limits of functions 151
Section 4.1: Limit of a function 151
Section 4.2: When something is not a limit 162
Section 4.3: Limit theorems 165
Section 4.4: Infinite limits (for real-valued functions) 173
Section 4.5: Limits at infinity 176
Chapter 5: Continuity 179
Section 5.1: Continuous functions 179
Section 5.2: Topology and the Extreme value theorem 184
Section 5.3: Intermediate Value Theorem 187
Section 5.4: Radical functions 190
Section 5.5: Uniform continuity 195
Chapter 6: Differentiation 198
Section 6.1: Definition of derivatives 198
Section 6.2: Basic properties of derivatives 202
Section 6.3: The Mean Value Theorem 210
Section 6.4: L’Hˆopital’s rule 213
Section 6.5: Higher-order derivatives, Taylor polynomials 217
Chapter 7: Integration 222
Section 7.1: Approximating areas 222
Section 7.2: Computing integrals from upper and lower sums 231
Section 7.3: What functions are integrable? 234
Section 7.4: The Fundamental Theorem of Calculus 239
Section 7.5: Natural logarithm and the exponential functions 247
Section 7.6: Applications of integration 252
Chapter 8: Sequences 259
Section 8.1: Introduction to sequences 259
Section 8.2: Convergence of infinite sequences 263
Section 8.3: Divergence of infinite sequences and infinite limits 271
5
Section 8.4: Convergence theorems via epsilon-N proofs 275
Section 8.5: Convergence theorems via functions 280
Section 8.6: Bounded sequences, monotone sequences, ratio test 284
Section 8.7: Cauchy sequences, completeness of R, C 287
Section 8.8: Subsequences 290
Section 8.9: Liminf, limsup for real-valued sequences 294
Chapter 9: Infinite series and power series 299
Section 9.1: Infinite series 299
Section 9.2: Convergence and divergence theorems for series 303
Section 9.3: Power series 309
Section 9.4: Differentiation of power series 314
Section 9.5: Numerical evaluations of some series 317
Section 9.6: Specialized uses of power series 318
Section 9.7: Taylor series 322
Section 9.8: A special function 326
Section 9.9: A special function, continued 328
Section 9.10: Trigonometry 333
Section 9.11: Examples of L’Hˆopital’s rule 336
Section 9.12: Further exotic uses of trigonometry 337
Appendix A: Advice on writing mathematics 343
Appendix B: What you should never forget 347
Index 351

Preface
These notes were writtenexpressly for Mathematics112 at Reed College, with first us-
age in the spring of 2013. The title of the course is “Introduction to Analysis”, prerequisite
iscalculus. RecentlyusedtextbookshavebeenRayMayer’sin-housenotes“Introductionto
Analysis”(2006,availableathttp://www.reed.edu/~mayer/math112.html/index.html),
and Steven R. Lay’s “Analysis, With an Introduction to Proof” (Prentice Hall, Inc., En-
glewood Cliffs, NJ, 1986, 4th edition).
In Math 112, students learn to write proofs while at the same time learning about bi-
nary operations, orders, fields, ordered fields, complete fields, complex numbers, sequences,
and series. We also review limits, continuity, differentiation, and integration. My aim for
these notes is to constitute a self-contained book that covers the standard topics of a
course in introductory analysis, that handles complex-valued functions, sequences, and se-
ries, that has enough examples and exercises, that is rigorous, and is accessible to Reed
College undergraduates.*
Chapter 1 is about how we do mathematics: basic logic, proof methods, and Pascal’s
triangle for practicing proofs. Chapter 2 introduces foundational concepts: sets, Carte-
sian products, relations, functions, binary operations, fields, ordered fields, Archimedean
property for the set of real numbers. In Chapters 1 and 2 we assume knowledge of high
school mathematics so that we do not practice abstract concepts and methods in a vac-
uum. Chapter 3 throughSection3.8takes a stepback: we“forget”most previously learned
mathematics, and we use the newly learned abstract tools to construct natural numbers,
integers, rational numbers, real numbers, with all the arithmetic and order. I do not teach
theseconstructionsingreatdetail;myaimistogiveasenseofthemandtopracticeabstract
logical thinking. The remaining sections in Chapter 3 are new material for most students:
the field of complex numbers, and some topology. I cover the last section of Chapter 3 very
lightly. Subsequent chapters cover standard material for introduction to analysis: limits,
continuity, differentiation, integration, sequences, series, ending with the development of
the power series ∞ xk, the exponential and the trigonometric functions. Since students
k=0 k!
have seen limits, continuity, differentiation and integration before, I go through chapters 4
P
through 7 quickly. I slow down for sequences and series (chapters 8 and 9).
An effortismadethroughouttouse onlywhathadbeenproved. Inparticular,trigono-
metric functions appear properly only in the last chapter, with occasional appearances in
theexercisesearlier(withcareful listingoftheneeded assumptionsfor thesefunctions). For
* I currently maintain two versions of these notes, one in which the natural, rational and real numbers are
constructed and the Least upper bound theorem is proved for the ordered field of real numbers, and one version
in which the Least upper bound property is assumed for the ordered field of real numbers. See my Math 112
webpage for links to the two versions.
www.reed.edu/~iswanson/112.html
8 Preface
this reason, the chapters on differentiation and integration do not have the usual palette of
examples of other books where differentiability and derivatives of trigonometric functions
are assumed.
I acknowledge and thank the support from the Dean of Faculty of Reed College to
fund exercise and proofreading support in the summer of 2012 for Maddie Brandt, Munyo
Frey-Edwards, and Kelsey Houston-Edwards. I further thank Mark Angeles, Josie Baker,
Anji Bodony, Zachary Campbell, Safia Chettih, Laura Dallago, Andrew Erlanger, Palak
Jain, Ya Jiang, Albyn Jones, Mason Kennedy, Christopher Keane, Michael Keppler, Oleks
Lushchyk, Molly Maguire, Benjamin Morrison, Samuel Olson, Kyle Ormsby, Ang´elica
Osorno, Shannon Pearson, David Perkinson, Jeremy Rachels, Marika Swanberg, Simon
Swanson, Matyas Szabo, Emerson Webb, Livia Xu, Qiaoyu Yang, and Dean Young for
their feedback. If you have further comments or corrections, please send them to iswan-
[email protected].
The briefest overview, motivation, notation
What are the meanings of the following:
5+6
7 9
·
8 4
−
4/5
√2
4 8
−
1/3 = 0.333... 1 = 3 1/3 = 0.999...
·
(a b)2 = a2 b2
· ·
(a+b) (c+d) = ac+ad+bc+bd
·
(a+b) (a b) = a2 b2
· − −
(a+b)2 = a2 +2ab+b2
√a √b = √ab (for which a,b?)
·
What is going on: √ 4 √ 9 = ( 4)( 9) = √36 = 6,
− · − − −
√ 4 √ 9 = 2i 3i =p6
− · − · −
You know all of the above except possibly the complex numbers in the last two rows,
where obviously something went wrong. We will not resolve this last issue until later in
the semester, but the point for now is that we do need to reason carefully.
The main goal of this class is to learn to reason carefully, rigorously. Since one cannot
reason in a vacuum, we will (but of course) be learning a lot of mathematics as well:
sets, logic, various number systems, fields, the field of real numbers, the field of complex
numbers, sequences, series, some calculus, and that eix = cosx+isinx.
We will make it all rigorous, i.e., we will be doing proofs. A proof is a sequence of
steps that logically follow from previously accepted knowledge.
But no matter what you do, never divide by 0. For further wise advice, turn to
Appendix A.
[Notational convention: Text between square brackets in this font and in
red color should be read as a possible reasoning going on in the background
in your head, and not as part of formal writing.]
1 Exercises with a dagger are invoked later in the text.
†
*2 Exercises with a star are more difficult.