Expert answer:I need help with real analysis mathematics

Answer & Explanation:exercises : 7.4.2 ,7.4.3 ,7.4.6 ,7.4.8 ,7.4.9 , 9.4.13 ,7.4.14  from chapter 7 in this the book in basic analysis basic analysis (real ).pdf
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Basic Analysis
Introduction to Real Analysis
by Jiří Lebl
December 16, 2014
2
Typeset in LATEX.
Copyright c 2009–2014 Jiří Lebl
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300, San Francisco, California, 94105, USA.
You can use, print, duplicate, share these notes as much as you want. You can base your own notes
on these and reuse parts if you keep the license the same. If you plan to use these commercially (sell
them for more than just duplicating cost), then you need to contact me and we will work something
out. If you are printing a course pack for your students, then it is fine if the duplication service is
charging a fee for printing and selling the printed copy. I consider that duplicating cost.
During the writing of these notes, the author was in part supported by NSF grants DMS-0900885
and DMS-1362337.
See http://www.jirka.org/ra/ for more information (including contact information).
Contents
Introduction
0.1 About this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2 About analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.3 Basic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
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Real Numbers
1.1 Basic properties . . . . . . . . . .
1.2 The set of real numbers . . . . . .
1.3 Absolute value . . . . . . . . . .
1.4 Intervals and the size of R . . . .
1.5 Decimal representation of the reals
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21
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Sequences and Series
2.1 Sequences and limits . . . . . . . . . . . . . . . . . .
2.2 Facts about limits of sequences . . . . . . . . . . . . .
2.3 Limit superior, limit inferior, and Bolzano-Weierstrass
2.4 Cauchy sequences . . . . . . . . . . . . . . . . . . . .
2.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 More on series . . . . . . . . . . . . . . . . . . . . . .
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43
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Continuous Functions
3.1 Limits of functions . . . . . . . . . . . .
3.2 Continuous functions . . . . . . . . . . .
3.3 Min-max and intermediate value theorems
3.4 Uniform continuity . . . . . . . . . . . .
3.5 Limits at infinity . . . . . . . . . . . . .
3.6 Monotone functions and continuity . . . .
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The Derivative
129
4.1 The derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.2 Mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3
4
CONTENTS
4.3
4.4
5
Taylor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Inverse function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
The Riemann Integral
5.1 The Riemann integral . . . . . . .
5.2 Properties of the integral . . . . .
5.3 Fundamental theorem of calculus .
5.4 The logarithm and the exponential
5.5 Improper integrals . . . . . . . . .
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147
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176
6
Sequences of Functions
189
6.1 Pointwise and uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.2 Interchange of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.3 Picard’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7
Metric Spaces
7.1 Metric spaces . . . . . . . . . . . . . . . . . .
7.2 Open and closed sets . . . . . . . . . . . . . .
7.3 Sequences and convergence . . . . . . . . . . .
7.4 Completeness and compactness . . . . . . . . .
7.5 Continuous functions . . . . . . . . . . . . . .
7.6 Fixed point theorem and Picard’s theorem again
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207
207
214
221
225
230
234
Further Reading
237
Index
239
Introduction
0.1
About this book
This book is a one semester course in basic analysis. It started its life as my lecture notes for
teaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in Fall semester 2009.
Later I added the metric space chapter to teach Math 521 at University of Wisconsin–Madison
(UW). A prerequisite for this course is a basic proof course, using for example [H], [F], or [DW].
It should be possible to use the book for both a basic course for students who do not necessarily
wish to go to graduate school (such as UIUC 444), but also as a more advanced one-semester course
that also covers topics such as metric spaces (such as UW 521). Here are my suggestions for what
to cover in a semester course. For a slower course such as UIUC 444:
§0.3, §1.1–§1.4, §2.1–§2.5, §3.1–§3.4, §4.1–§4.2, §5.1–§5.3, §6.1–§6.3
For a more rigorous course covering metric spaces that runs quite a bit faster (such as UW 521):
§0.3, §1.1–§1.4, §2.1–§2.5, §3.1–§3.4, §4.1–§4.2, §5.1–§5.3, §6.1–§6.2, §7.1–§7.6
It should also be possible to run a faster course without metric spaces covering all sections of
chapters 0 through 6. The approximate number of lectures given in the section notes through chapter
6 are a very rough estimate and were designed for the slower course. The first few chapters of the
book can be used in an introductory proofs course as is for example done at Iowa State University
Math 201, where this book is used in conjunction with Hammack’s Book of Proof [H].
The book normally used for the class at UIUC is Bartle and Sherbert, Introduction to Real
Analysis third edition [BS]. The structure of the beginning of the book somewhat follows the
standard syllabus of UIUC Math 444 and therefore has some similarities with [BS]. A major
difference is that we define the Riemann integral using Darboux sums and not tagged partitions.
The Darboux approach is far more appropriate for a course of this level.
Our approach allows us to fit a course such as UIUC 444 within a semester and still spend
some extra time on the interchange of limits and end with Picard’s theorem on the existence and
uniqueness of solutions of ordinary differential equations. This theorem is a wonderful example
that uses many results proved in the book. For more advanced students, material may be covered
faster so that we arrive at metric spaces and prove Picard’s theorem using the fixed point theorem as
is usual.
5
6
INTRODUCTION
Other excellent books exist. My favorite is Rudin’s excellent Principles of Mathematical
Analysis [R2] or as it is commonly and lovingly called baby Rudin (to distinguish it from his
other great analysis textbook). I took a lot of inspiration and ideas from Rudin. However, Rudin
is a bit more advanced and ambitious than this present course. For those that wish to continue
mathematics, Rudin is a fine investment. An inexpensive and somewhat simpler alternative to Rudin
is Rosenlicht’s Introduction to Analysis [R1]. There is also the freely downloadable Introduction to
Real Analysis by William Trench [T].
A note about the style of some of the proofs: Many proofs traditionally done by contradiction,
I prefer to do by a direct proof or by contrapositive. While the book does include proofs by
contradiction, I only do so when the contrapositive statement seemed too awkward, or when
contradiction follows rather quickly. In my opinion, contradiction is more likely to get beginning
students into trouble, as we are talking about objects that do not exist.
I try to avoid unnecessary formalism where it is unhelpful. Furthermore, the proofs and the
language get slightly less formal as we progress through the book, as more and more details are left
out to avoid clutter.
As a general rule, I use := instead of = to define an object rather than to simply show equality.
I use this symbol rather more liberally than is usual for emphasis. I use it even when the context is
“local,” that is, I may simply define a function f (x) := x2 for a single exercise or example.
Finally, I would like to acknowledge Jana Maříková, Glen Pugh, Paul Vojta, Frank Beatrous,
and Sönmez Şahutoğlu for teaching with the book and giving me lots of useful feedback. Frank
Beatrous wrote the University of Pittsburgh version extensions, which served as inspiration for
many of the recent additions. I would also like to thank Dan Stoneham, Jeremy Sutter, Eliya Gwetta,
Daniel Alarcon, Steve Hoerning, Yi Zhang, Nicole Caviris, Kenji Kozai, Kristopher Lee, Baoyue Bi,
Hannah Lund, an anonymous reader, and in general all the students in my classes for suggestions
and finding errors and typos.
0.2. ABOUT ANALYSIS
0.2
7
About analysis
Analysis is the branch of mathematics that deals with inequalities and limits. The present course
deals with the most basic concepts in analysis. The goal of the course is to acquaint the reader with
rigorous proofs in analysis and also to set a firm foundation for calculus of one variable.
Calculus has prepared you, the student, for using mathematics without telling you why what
you learned is true. To use, or teach, mathematics effectively, you cannot simply know what is true,
you must know why it is true. This course shows you why calculus is true. It is here to give you a
good understanding of the concept of a limit, the derivative, and the integral.
Let us use an analogy. An auto mechanic that has learned to change the oil, fix broken
headlights, and charge the battery, will only be able to do those simple tasks. He will be unable to
work independently to diagnose and fix problems. A high school teacher that does not understand
the definition of the Riemann integral or the derivative may not be able to properly answer all the
students’ questions. To this day I remember several nonsensical statements I heard from my calculus
teacher in high school, who simply did not understand the concept of the limit, though he could “do”
all problems in calculus.
We start with a discussion of the real number system, most importantly its completeness property,
which is the basis for all that comes after. We then discuss the simplest form of a limit, the limit of
a sequence. Afterwards, we study functions of one variable, continuity, and the derivative. Next, we
define the Riemann integral and prove the fundamental theorem of calculus. We discuss sequences
of functions and the interchange of limits. Finally, we give an introduction to metric spaces.
Let us give the most important difference between analysis and algebra. In algebra, we prove
equalities directly; we prove that an object, a number perhaps, is equal to another object. In analysis,
we usually prove inequalities. To illustrate the point, consider the following statement.
Let x be a real number. If 0 ≤ x < ε is true for all real numbers ε > 0, then x = 0.
This statement is the general idea of what we do in analysis. If we wish to show that x = 0, we
show that 0 ≤ x < ε for all positive ε. The term real analysis is a little bit of a misnomer. I prefer to use simply analysis. The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. Furthermore, a more advanced course on real analysis would talk about complex numbers often. I suspect the nomenclature is historical baggage. Let us get on with the show. . . 8 INTRODUCTION 0.3 Basic set theory Note: 1–3 lectures (some material can be skipped or covered lightly) Before we start talking about analysis we need to fix some language. Modern∗ analysis uses the language of sets, and therefore that is where we start. We talk about sets in a rather informal way, using the so-called “naïve set theory.” Do not worry, that is what majority of mathematicians use, and it is hard to get into trouble. We assume the reader has seen basic set theory and has had a course in basic proof writing. This section should be thought of as a refresher. 0.3.1 Sets Definition 0.3.1. A set is a collection of objects called elements or members. A set with no objects is called the empty set and is denoted by 0/ (or sometimes by {}). Think of a set as a club with a certain membership. For example, the students who play chess are members of the chess club. However, do not take the analogy too far. A set is only defined by the members that form the set; two sets that have the same members are the same set. Most of the time we will consider sets of numbers. For example, the set S := {0, 1, 2} is the set containing the three elements 0, 1, and 2. We write 1∈S to denote that the number 1 belongs to the set S. That is, 1 is a member of S. Similarly we write 7∈ /S to denote that the number 7 is not in S. That is, 7 is not a member of S. The elements of all sets under consideration come from some set we call the universe. For simplicity, we often consider the universe to be the set that contains only the elements we are interested in. The universe is generally understood from context and is not explicitly mentioned. In this course, our universe will most often be the set of real numbers. While the elements of a set are often numbers, other object, such as other sets, can be elements of a set. A set may contain some of the same elements as another set. For example, T := {0, 2} contains the numbers 0 and 2. In this case all elements of T also belong to S. We write T ⊂ S. More formally we make the following definition. ∗ The term “modern” refers to late 19th century up to the present. 0.3. BASIC SET THEORY 9 Definition 0.3.2. (i) A set A is a subset of a set B if x ∈ A implies x ∈ B, and we write A ⊂ B. That is, all members of A are also members of B. (ii) Two sets A and B are equal if A ⊂ B and B ⊂ A. We write A = B. That is, A and B contain exactly the same elements. If it is not true that A and B are equal, then we write A 6= B. (iii) A set A is a proper subset of B if A ⊂ B and A 6= B. We write A ( B. When A = B, we consider A and B to just be two names for the same exact set. For example, for S and T defined above we have T ⊂ S, but T 6= S. So T is a proper subset of S. At this juncture, we also mention the set building notation, {x ∈ A : P(x)}. This notation refers to a subset of the set A containing all elements of A that satisfy the property P(x). The notation is sometimes abbreviated (A is not mentioned) when understood from context. Furthermore, x ∈ A is sometimes replaced with a formula to make the notation easier to read. Example 0.3.3: The following are sets including the standard notations. (i) The set of natural numbers, N := {1, 2, 3, . . .}. (ii) The set of integers, Z := {0, −1, 1, −2, 2, . . .}. (iii) The set of rational numbers, Q := { mn : m, n ∈ Z and n 6= 0}. (iv) The set of even natural numbers, {2m : m ∈ N}. (v) The set of real numbers, R. Note that N ⊂ Z ⊂ Q ⊂ R. There are many operations we want to do with sets. Definition 0.3.4. (i) A union of two sets A and B is defined as A ∪ B := {x : x ∈ A or x ∈ B}. (ii) An intersection of two sets A and B is defined as A ∩ B := {x : x ∈ A and x ∈ B}. (iii) A complement of B relative to A (or set-theoretic difference of A and B) is defined as A B := {x : x ∈ A and x ∈ / B}. 10 INTRODUCTION (iv) We say complement of B and write Bc if A is understood from context. The set A is either the entire universe or is the obvious set containing B. (v) We say sets A and B are disjoint if A ∩ B = 0. / The notation Bc may be a little vague at this point. If the set B is a subset of the real numbers R, then Bc means R B. If B is naturally a subset of the natural numbers, then Bc is N B. If ambiguity would ever arise, we will use the set difference notation A B. A B A A∪B A B A∩B B B Bc AB Figure 1: Venn diagrams of set operations. We illustrate the operations on the Venn diagrams in Figure 1. Let us now establish one of most basic theorems about sets and logic. Theorem 0.3.5 (DeMorgan). Let A, B,C be sets. Then (B ∪C)c = Bc ∩Cc , (B ∩C)c = Bc ∪Cc , or, more generally, A (B ∪C) = (A B) ∩ (A C), A (B ∩C) = (A B) ∪ (A C). 0.3. BASIC SET THEORY 11 Proof. The first statement is proved by the second statement if we assume the set A is our “universe.” Let us prove A (B ∪C) = (A B) ∩ (A C). Remember the definition of equality of sets. First, we must show that if x ∈ A (B ∪C), then x ∈ (A B) ∩ (A C). Second, we must also show that if x ∈ (A B) ∩ (A C), then x ∈ A (B ∪C). So let us assume x ∈ A (B ∪C). Then x is in A, but not in B nor C. Hence x is in A and not in B, that is, x ∈ A B. Similarly x ∈ A C. Thus x ∈ (A B) ∩ (A C). On the other hand suppose x ∈ (A B) ∩ (A C). In particular x ∈ (A B) and so x ∈ A and x ∈ / B. Also as x ∈ (A C), then x ∈ / C. Hence x ∈ A (B ∪C). The proof of the other equality is left as an exercise. We will also need to intersect or union several sets at once. If there are only finitely many, then we simply apply the union or intersection operation several times. However, suppose we have an infinite collection of sets (a set of sets) {A1 , A2 , A3 , . . .}. We define ∞ [ An := {x : x ∈ An for some n ∈ N}, n=1 ∞ An := {x : x ∈ An for all n ∈ N}. n=1 We can also have sets indexed by two integers. For example, we can have the set of sets {A1,1 , A1,2 , A2,1 , A1,3 , A2,2 , A3,1 , . . .}. Then we write ! ∞ [ ∞ [ n=1 m=1 An,m = ∞ [ ∞ [ n=1 m=1 An,m . And similarly with intersections. It is not hard to see that we can take the unions in any order. However, switching the order of unions and intersections is not generally permitted without proof. For example: ∞ ∞ [ {k ∈ N : mk < n} = ∞ [ 0/ = 0. / n=1 m=1 n=1 ∞ [ ∞ ∞ However, {k ∈ N : mk < n} = m=1 n=1 0.3.2 N = N. m=1 Induction When a statement includes an arbitrary natural number, a common method of proof is the principle of induction. We start with the set of natural numbers N = {1, 2, 3, . . .}, and we give them their natural ordering, that is, 1 < 2 < 3 < 4 < · · · . By S ⊂ N having a least element, we mean that there exists an x ∈ S, such that for every y ∈ S, we have x ≤ y. 12 INTRODUCTION The natural numbers N ordered in the natural way possess the so-called well ordering property. We take this property as an axiom; we simply assume it is true. Well ordering property of N. Every nonempty subset of N has a least (smallest) element. The principle of induction is the following theorem, which is equivalent to the well ordering property of the natural numbers. Theorem 0.3.6 (Principle of induction). Let P(n) be a statement depending on a natural number n. Suppose that (i) (bas ... Purchase answer to see full attachment

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