# Andrew McLennan January 19, Winter Lecture 5. A. Two of the most fundamental notions of the dierential calculus (recall that

Save this PDF as:

Size: px
Start display at page:

Download "Andrew McLennan January 19, Winter Lecture 5. A. Two of the most fundamental notions of the dierential calculus (recall that"

## Transcription

1 Andrew McLennan January 19, 1999 Economics 5113 Introduction to Mathematical Economics Winter 1999 Lecture 5 Convergence, Continuity, Compactness I. Introduction A. Two of the most fundamental notions of the dierential calculus (recall that Leibniz and Newton are credited with being the creators of this) `limit' and `continuity,' were not successfully described with formal precision until the nineteenth century. 1. After a period of evolution, the mathematics profession settled on two principle frameworks, metric spaces and topological spaces, as the arenas in which these concepts are dened and discussed. 2. Let X be a set. A function d : X X! IR + is a metric if: a. for all x; y 2 X, d(x; y) = 0 if and only if x = y; b. for all x; y 2 X, d(x; y) =d(y; x); c. for all x; y; z 2 X, d(x; z) d(x; y) +d(y; z). A metric space is a pair (X; d) where X is a set and d is a metric on X. 3. The most fundamental geometric objects in X are the open balls: for x 2 X and >0, let B (x) =fx 0 2X:d(x; x 0 ) <g: 1

2 II. Sequences and Convergence A. Formally,asequence inxis a function from f1; 2; 3;:::g, or some similar index set, tox. 1. Informally, a sequence is akin to a set, and we denote it by fx n g (or sometimes x 1 ;x 2 ; :::) suggesting that we are concerned less with the particular ordering and more with the general tendency as n gets large. B. TRoughly speaking, the sequence fx n g converges to the limit x 2 X if it is eventually inside each B (x). 1. In more detail, for any >0 there is an integer N such that d(x n ;x) < for all n N. 2. The notation x n! x indicates that the sequence fx n g converges to x. 3. A sequence that converges to something is said to be convergent. Sequences that are not convergent are said to be divergent. 4. There are two ways that a sequence might be divergent: a. The sequence might just bounce around without ever settling down. This happens when, for some >0, it is possible, for any integer N, tondn; m N, such that d(x n ;x m ). b. A Cauchy sequence is a sequence such that for any >0there is an integer N such that d(x n ;x m ) <for all n; m N, i.e., a sequence that does not bounce around too much. c. If a Cauchy sequence fails to converge, the usual view is that this is not because the sequence is in any sense ill behaved. Instead, the fault lies with the given space. A metric space is said to be complete if every Cauchy sequence has a limit. C. Sequences in IR. 1. The basic reason that the real numbers are more useful than the rational numbers is as follows: 2

3 Theorem: IR is complete. Proof: Let ft n g be a Cauchy sequence in IR. Let S be the set of real numbers S such that t n >sfor all but at most nitely many n. It is straightforward to use the denition of a Cauchy sequence to show that S is nonempty and bounded above. Let s be the least uppoer bound of S. We claim that t n! s. In order to prove this by contradiction we suppose that it is not the case, which means that there is some >0 such that jt n, sj > for innitely many n. Suppose N is large enough that jt m, t n j <=2 for all n; m N, and choose M N such that jt M, sj >. Then jt n, t M j <=2 for all n M, soif t M >s+=2, then s + =2 is an element ofs, while if t M <s,=2, then s, =2 is an upper bound. In either case we have contradicted the assumption that s is the least uppoer bound of S. 2 It turns out that any metric space X can be \completed" by adding \the missing points." The formal procedure for doing this is to let X be the set of equivalence classes of Cauchy sequences in X, where two Cauchy sequences are equivalent if the distance between them goes toi zero in the limit. The details (dening the natural metric on X and showing that X is complete) are lengthy, so I will not discuss them, but that is not to say that it would be bad for you to think carefully about what is involved. 3 A sequence ft k )g in IR is increasing if t k t k+1 for all k, and it is strictly increasing if t k <t k+1 for all k. The sequence ft k g is bounded above if there is some t such that t k t for all k. Theorem: An increasing sequence ft k )g that is bounded above has a limit. Proof: We leave this as an exercise, since the ideas are very similar to those used to prove the last result. C. Sequences in IR n. 1. Unless we explicitly say otherwise, IR n is always endowed with the Eu- 3

4 clidean metric: d(x; y) = kx,yk. 2. Basically all you need to know about convergence in IR n is: Theorem: A sequence fx k =(x 1 k ;:::;xn k )gin IRn converges if and only if each component sequence fx i kg is convergent. Proof: If the sequence converges, say to x, then each component sequence converges because, for all i =1;:::n,jx i,x i k jkx,x kk. Conversely, suppose that each component sequence fx i k g converges to xi, and dene x to be the point (x 1 ;:::;x n ). For any y 2 IR n we have Consequently kyk 2 =jy 1 j 2 +:::+jy n j 2 (jy 1 j + :::+jy n j) 2 : kx k, xk jx 1 k,x 1 j+:::+jx n k,x 1 j!0: III. Open and Closed Sets A. A set C X is closed (or closed inxif some other containing space is possible) if it contains all its limit points. That is, whenever fx k g is a sequence in C that converges to some point x, x is an element ofc. 1. Example: IR n + is closed. B. A neighborhood of a point x 2 X is any set S X that contains B (x) for some >0. A set U X is open (or open in X) if it is a neighborhood of each of its points, so tha t for each x 2 U there is >0 for which B (x) U. 1. Example: IR n ++ is open. Theorem: For any x 2 X and >0, B (x) is open. Proof: For any y 2 B (x), the triangle inequality implies that B (,d(x;y)) (y) B (x). 3. Exercise: Prove that, for any x and, f y 2 X : d(x; y) >gis open. Theorem: A set U X is open if and only if its complement U c = XnU is closed. 4

5 Proof: Let C = U c. The assertion consists of two implications, the `if' and the `only if.' Suppose that C is closed, and x 2 U. If, for each natural number k, B 1=k (x) 6 U, we can choose a point inb 1=k (x) \ C, thereby constructing a sequence fx k g in C that converges to x. Since C is closed, this w ould imply that x 2 C, contrary to our assumption that x 2 U. Therefore B 1=k (x) U for large k, and since x was an arbitrary point ofu, wehave shown that U is open. Suppose that U is open, and that fx k g is a sequence in C that converges to x. If x 2 U, then B (x) U for some >0, and x k 2 B (x) for large k since x k! x, but this contradicts the a ssumption that x k 2 C. Thus C is closed. C. A topological space is a pair (T;) in which T is a set and is a collection of subsets of T, called the open sets of T, with the properties of the open subsets of X given by: Theorem: (a) ; and X are open sets. (b) The intersection of nitely many open sets is open. (c) The union of an arbitrary collection of open sets is open. Proof: This is all pretty obvious, so we will only mention that (b) is proved by noting that if U 1 ;:::;U p are open and B 1 (x) U 1 ;:::;B p (x) U p, then B minf1 ;::: p g(x) U 1 \ :::\U p : 1. Exercise: In a topological space a closed set is by denition a set whose complement is open. The collection of all closed subsets of T has properties that are similar to, and immediately derivable from, (a){(c). What are they? 2. The theory of topological spaces is much more complicated than the theory of metric spaces, essentially because any number of things can 5

6 go wrong. For starters, in a topological space there can be sets that contain all the limits of their convergent sequences, but are nonetheless not closed. 3. In the future, at least, I will try to be careful to give denitions that are valid for all topological spaces, not just metric spaces, and to be careful to indicate what properties of metric spaces might not be true more generally. However, we are basically going to just forget about general topological spaces. IV. Continuity A. Let (X; d X ) and (Y;d Y ) be metric spaces, and let f : X! Y be a function. 1. The function f is continuous if f,1 (V ):=fx2x:f(x)2vgis open whenever V Y is open. 2. This denition makes sense, and is correct, for general topological spaces, but has the unfortunate aspect of seeming strange on rst sight. Note, however, that if f is continuous, x 2 X, and >0, then f,1 (B (f(x))) is open, so that there exists some >0 such that B (x) f,1 (B (f(x))). 3. Exercise: The reverse implication also holds: (8x 2 X)(8 >0)(9 >0) B (x) f,1 (B (f(x))) implies that f is continuous. Prove this. B. The following test of continuity for functions between metric spaces is very useful. Theorem: f is continuous if and only if f(x n )! f(x) whenever fx n g is a sequence in X that converges to x. Proof: Suppose f is continuous. Let fx n g be a sequence in X with x n! x. For any > 0, f,1 (B (f(x))) is open and contains x, so there is some > 0 such that 6

7 B (x) f,1 (B (f(x))). Since x n! x, for all suciently large n we havex n 2B (x) and f(x n ) 2 B (f(x)). Since was arbitrary, wehave shown that f(x n )! f(x). Suppose that f(x n )! f(x) whenever x n! x. Let V Y be open. If f,1 (V )is not open, it must contain a point x such that for each natural number n we can choose x n 2 B 1=n (x) n f,1 (V ). Then x n! x. But V is open, so there is > 0 such that B (f(x)) V. For each n, f(x n ) =2 V, so that d(f(x n );f(x)) >and f(x n ) 6! f(x), contrary to assumption. Therefore f,1 (V ) is open, and since V was arbitrary, wehave shown that f is continuous. V. Compactness A. The concept of a compact set, developed in the rst part of this century, is now applied in most aspects of mathematics. B. The denition is far from intuitive. Let (X; d) be a metric space. 1. An open cover of a set C X is a collection fu g 2A of open sets that \covers" C in the sense that C 0 [ 2A U. 2. A subcover is a subset of fu g 2A that also covers C. 3 This set C is compact if every open cover has a nite subcover. C. It is easy to explain why such sets might be attractive from the point of view of optimization. Theorem: If f: X! IR is continuous, and C X is compact, then f attains its supremum on C (that is, arg max x2c f(x) is nonempty). Proof: For each x 2 C let U x = fy 2 X: f(y) <f(x)g=f,1 (,1;f(x)) Since f is continuous, U x is open. If f does not attain its maximum on C, then every point ofcis in some U x so fu x g x2c is a cover of C. Let U x1 ;:::;U xk be a nite subcover. 7

8 Reordering if we need to, suppose that f(x 1 ) f(x 2 ) f(x k ). Then U x1 U x2 U xk,sowehave CU x1 [[U xk = U xk : Since x k 2 C, this would yield f(x k ) <f(x k ). This contradiction completes the proof. D. What kinds of sets are compact? 1. Finite sets are obviously compact. Other examples are not so obvious. 2. A set C X is bounded if, for any x 2 C, there is M > 0 such that d(x; x 0 ) M for all x 0 2 C. Lemma: If C X is compact, it is bounded. Proof: For any x 2 C, the sets U k = fx 2 C: d(x; y) <kg (k=1;2;:::) constitute an open cover of C, and must have a nite subcover. Lemma: If C X is compact, it is closed. Proof: Suppose fx n g is a sequence in C that converges to x. Ifx=2C, the sets U k = y 2 C: d(x; y) > 1 (k =1;2;:::) k constitute an open cover of C which cannot have a nite subcover if there is a sequence in C converging to x. 3. A subsequence of a sequence fx n g is a sequence x n1 ;x n2 ;::: where n 1 < n 2 <. 4. An accumulation point of a sequence fx n g is a point x with the property that, for any >0, there are innitely many n such that x n 2 B (x). 8

9 a. Exercise: Prove that a sequence fx n g has a convergent subsequence if and only if it has an accumulation point. 5. A set C X is sequentially compact is every sequence in C has a convergent subsequence whose limit is in C. Theorem: A compact set C X is sequentially compact. Proof: By the exercise, it suces to show that a given sequence fx k g in C has an accumulation point. If not, for each x it is possible to nd an open set U x that contains x k for at most nitely many k. Since x 2 U x, for each x, fu x : x 2 Cg is an open cover of C. But the union of the elements of a nite subcover contains C, and contains x k for at most nitely many k. (The sum of nitely many nite numbers is nite.) Since this is impossible, the proof is complete. 6. The converse is also true for metric spaces (we will prove this shortly) but not for general topological spaces. E. Compact Subsets of Euclidean Space. Lemma: If fx k g is a bounded sequence in IR n, it has a convergent subsequence. Proof: We claim that it suces to prove this in the case n =1.Ifitisknown to be true for n = 1, then, in the general case, we can choose a subsequence such that the sequence of rst components is convergent, take a further subsequence of this subsequence for which the sequence of record components converges, and so on until, in the nth sequence, all sequences of components are convergent. From an earlier exercise we know that this is sucient for convergence. So, let ft k g be a bounded sequence in IR, with lower bound a 0 and upper bound b 0. We construct a sequence [a 1 ;b 1 ];[a 2 ;b 2 ];::: of closed intervals \inductively" by letting [a i ;b i ]= a i,1 ; a i,1+b i,1 2 9

10 if the right hand side contains t k for innitely many k. Otherwise we set [a i ;b i ]= ai,1 +b i,1 ;b i : 2 Arguing by induction, we can easily see that each [a i ;b i ] contains t k for innitely many k, since this is true for [a i,1 ;b i,1 ]. Now choose k 1 such that t k1 2 [a 1 ;b 1 ]. We construct fk i g inductively by choosing k i >k i,1 with t k 2 [a i ;b i ]. Since t ki ;t ki+1 ;::: are all contained in [a i ;b i ], which has length (b 0,a 0 ) 2 i, ft ki g is a Cauchy sequence, hence convergent since IR is complete. 1. All that remains is to bundle our ndings in a nice neat package. Theorem: A set C IR n is compact if and only if it is closed and bounded. Proof: We have already shown that a compact C is closed and bounded. Suppose that C is closed and bounded. Consider that any sequence in C has a convergent subsequence, by the limit, and the limit must be in C since C is closed. Thus C is sequentially compact, so it is compact since IR n is a metric space. V. Countable and Uncountable Sets A. Two sets X and Y have the same cardinality if there is a bijection f : X! Y. B. A set is said to be countable if it has the same cardinality as the natural numbers N := f1; 2; 3;:::g. 1. Some authors say that a set is countable if it is either nite or has the same cardinality asn, using the phrase \countably innite" to describe a set with exactly the same cardinality asn. 2. A set that is neither nite nor countable is said to be uncountable. C. Properties of Countable Sets. 1. Any innite subset of a countable set is countable. (Pf: If Y X = fx 1 ;x 2 ;:::g with Y innite, we map Y bijectively to N by letting f(x n ) be the number of i n such that x i 2 Y.) 10

11 2. The cartesian product of two countable sets is countable. 3. Examples: a. The rational numbers are countable since they can be put in one to one correspondence with a subset of N N. b. The real numbers are not countable. This is proved by producing a contradiction using Cantor's diagonalization procedure: if r 1 ;r 2 ;::: is an enumeration of the reals, create a number between 0 and 1 by choosing a rst digit (after the decimal point) that is dierent from the rst digit of r 1, a second digit dierent from the second digit of r 2, and so forth. All chosen digits can be dierent from 0 and 9, to avoid ambiguity about numbers such as 1 that may be written either 1:000 ::: of 0:999 :::.) The number constructed in this way is dierent from any number in the list. D. Let fu g 2A be a collection of open sets. Another collection of open sets fv g 2B is a renement of fu g 2A if S 2B V = S 2A U and, for each 2 B, there is some 2 A such that V U. Lemma: Any collection of fu g 2A of open subsets of IR n has a countable renement. Proof: Let B be the set of pairs (x; r) 2 IR n (0; 1) such that r and all components of x are rational numbers, and V (x;r) := B r (x) U for some. Then B is countable, by virtue of our remarks above, and we only need to show that S (x;r)2b V (x;r) = S 2A U. Choose 2 A and y 2 U arbitrarily. Then B (y) U for some >0. Choose a point x with rational coordinates in B =2 (y), and let r be a rational number between kx, yk and,kx,yk. Then y 2 B r (x) B (y) U : Lemma: If fv g 2B is a renement offu g 2A, and for some set S, nitely many elements of fv g 2B cover S, then there is a nite subcover of fu g 2A that covers S. 11

12 Proof: If V 1 ;:::;V K covers S, and for each k =1;:::;K, V K U k, then U 1 ;:::;U K covers S. Theorem: If C IR n is sequentially compact, then it is compact. Proof: Let fu g 2A be an open cover of C. This cover has a countable renement, and it suces to nd a nite subcover of the renement, which means that we only need to prove the result when A is countable, so we may assume that A = N. That is, we assume that the open cover is of the form fu 1 ;U 2 ;:::g. If there is no nite subcover, then for each k =1;2;::: wemaychoose x k 2 C n (U 1 [ :::[U k,1 ): By assumption fx k g 1 k=1 has a convergent subsequence whose limit x is in C. There is some K such that x 2 U K, and there is some >0 such that B (x) U K,sokx k,xkfor all but nitely many k, which is impossible if fx k g 1 k=1 has a subsequence converging to x. This contradiction completes the proof. 12

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### Mathematics for Computer Science

Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### Domain representations of topological spaces

Theoretical Computer Science 247 (2000) 229 255 www.elsevier.com/locate/tcs Domain representations of topological spaces Jens Blanck University of Gavle, SE-801 76 Gavle, Sweden Received October 1997;

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Some other convex-valued selection theorems 147 (2) Every lower semicontinuous mapping F : X! IR such that for every x 2 X, F (x) is either convex and

PART B: RESULTS x1. CHARACTERIZATION OF NORMALITY- TYPE PROPERTIES 1. Some other convex-valued selection theorems In this section all multivalued mappings are assumed to have convex values in some Banach

### Math 104: Introduction to Analysis

Math 104: Introduction to Analysis Evan Chen UC Berkeley Notes for the course MATH 104, instructed by Charles Pugh. 1 1 August 29, 2013 Hard: #22 in Chapter 1. Consider a pile of sand principle. You wish

### Chapter 7. Continuity

Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this

### Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces

Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z

### CHAPTER 1 BASIC TOPOLOGY

CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Georg Cantor (1845-1918):

Georg Cantor (845-98): The man who tamed infinity lecture by Eric Schechter Associate Professor of Mathematics Vanderbilt University http://www.math.vanderbilt.edu/ schectex/ In papers of 873 and 874,

### {f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### 13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite

### n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)

Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Real Numbers and Monotone Sequences

Real Numbers and Monotone Sequences. Introduction. Real numbers. Mathematical analysis depends on the properties of the set R of real numbers, so we should begin by saying something about it. There are

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. Prerequisites

### LECTURE NOTES IN MEASURE THEORY. Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12)

1 LECTURE NOTES IN MEASURE THEORY Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12) 2 PREFACE These are lecture notes on integration theory for a eight-week

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction

ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity

### 1 Norms and Vector Spaces

008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

### 4. Expanding dynamical systems

4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

### REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

### Set theory as a foundation for mathematics

Set theory as a foundation for mathematics Waffle Mathcamp 2011 In school we are taught about numbers, but we never learn what numbers really are. We learn rules of arithmetic, but we never learn why these

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### 10.2 Series and Convergence

10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

### Mathematical Induction. Mary Barnes Sue Gordon

Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by

### Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### Open and Closed Sets

Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Sets and set operations

CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

### Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### INTRODUCTION TO TOPOLOGY

INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology

### 1 Fixed Point Iteration and Contraction Mapping Theorem

1 Fixed Point Iteration and Contraction Mapping Theorem Notation: For two sets A,B we write A B iff x A = x B. So A A is true. Some people use the notation instead. 1.1 Introduction Consider a function

### Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

### ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Worksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation

Worksheet on induction MA113 Calculus I Fall 2006 First, let us explain the use of for summation. The notation f(k) means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other

### Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.

Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### Section 6-2 Mathematical Induction

6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Chapter 12 Modal Decomposition of State-Space Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor

Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### 1. Let X and Y be normed spaces and let T B(X, Y ).

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

### PSEUDOARCS, PSEUDOCIRCLES, LAKES OF WADA AND GENERIC MAPS ON S 2

PSEUDOARCS, PSEUDOCIRCLES, LAKES OF WADA AND GENERIC MAPS ON S 2 Abstract. We prove a Bruckner-Garg type theorem for the fiber structure of a generic map from a continuum X into the unit interval I. We

ADVANCED CALCULUS Lecture notes for MA 440/540 & 441/541 2015/16 Rudi Weikard log x 1 1 2 3 4 5 x 1 2 Based on lecture notes by G. Stolz and G. Weinstein Version of September 3, 2016 1 Contents First things

### Recursion Theory in Set Theory

Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied

### Set theory as a foundation for mathematics

V I I I : Set theory as a foundation for mathematics This material is basically supplementary, and it was not covered in the course. In the first section we discuss the basic axioms of set theory and the

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts