Week 2: Sequences and Series
|
|
- Arron Francis
- 5 years ago
- Views:
Transcription
1 QF0: Quantitative Finance August 29, 207 Week 2: Sequences and Series Facilitator: Christopher Ting AY 207/208 Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate. Leonhard Euler 2. What is a sequence?. Let A be a set of objects. A sequence {a i } i= of elements of A is a function from the set of natural numbers N to the set A, i.e. a correspondence that associates one and only one element of A to each natural number n N. In other words, a sequence of elements of A is an ordered list of elements of A, where the ordering is provided by the natural numbers. 2. Thus if {a i } i= is a sequence, a is the first element, a 2 is the second element, and so on. Example: Define a sequence {a i } i= by characterizing its n-th element a n as a n = n. So it is a sequence of rational numbers: a =, a 2 = /2,.... Example: Define a sequence {a i } i= where each a n is a closed interval of real numbers, a n = [ n +, ], n which is a set of real numbers that are bounded by [ and inclusive ] [ of /(n + ) and /n. The elements of this sequence are closed intervals: a = 2,, a 2 = 3, ], and so on. 2 Example: Fibonacci sequence The Fibonacci sequence comprises the numbers in the following integer sequence: 0,,, 2, 3, 5, 8, 3, 2, 34, 55, 89, This sequence is generated by the recurrence relation, with F 0 = 0 and F = : F n = F n + F n 2. 2-
2 Week 2: Sequences and Series Countable and uncountable sets. Let A be a set of objects. A is a countable set if all its elements can be arranged into a sequence, i.e. if there exists a sequence {a i } i= such that a A, n N : a n = a. In other words, A is a countable set if there exists at least one sequence {a i } i= such that every element of A belongs to the sequence. A is an uncountable set if such a sequence does not exist. The most important example of an uncountable set is the set of real numbers R. Example: The set of rational numbers Q is countable. Example: The set of irrational numbers is uncountable. 2. An uncountable sets are so densely populated that make counting impossible. 3. In mathematics, the cardinality of a set A is a measure of the number of elements of the set, and it is denoted by A. The cardinality of the natural numbers is denoted by ℵ A set A is uncountable if the cardinality of A is neither finite nor equal to ℵ 0, i.e., A, and A ℵ Limit of a sequence. First, we define the distance between two real numbers. It is the absolute value of their difference. For example, if a R and a n is a term of a sequence {a n }, the distance between a n and a, denoted by d(a n, a) is d(a n, a) = a n a. 2. Let {a n } be a sequence of real numbers. Let n 0 N. Denote by {a n } n>n0 a subsequence (subset of a sequence) of {a n } obtained by dropping the first n 0 terms, i.e., {a n } n>n0 = { a n0 +, a n0 +2,..., }. 3. Let a be a real number. We say that a is a limit of a sequence {a n } of real numbers, if, by appropriately choosing n 0, the distance between a and any term of the subsequence {a n } n>n0 can be made as close to zero as we like.
3 Week 2: Sequences and Series More formally, let a R. We say that a is a limit of a sequence {a n } of real numbers, if ɛ > 0, n 0 N : d(a n, a) < ɛ, n > n 0. Namely, for any arbitrarily small number ɛ, there exists a natural number n 0 such that the distance between a n and a is less than ɛ for all the terms a n with n > n 0. In other words, for any arbitrarily small number ɛ, one can find a subsequence {a n } n>n0 such that the distance between any term of the subsequence and a is less than ɛ. By dropping a sufficiently high number of initial terms of {a n }, one can make the remaining terms as close to a as one wishes. 5. If a is a limit of the sequence {a n }, we say that the sequence {a n } is a convergent sequence and that it converges to a. We indicate the fact that a is a limit of {a n } by a = lim n a n. Example: Define a sequence {a n } by characterizing its n-th element a n as follows: a n = n. Intuitively, the limit of the sequence should be 0: 0 = lim n a n. To prove it, choose any ɛ > 0 and find an n 0 N such that all terms of the subsequence {a n } n>n0 have distance from zero less than ɛ: d(a n, 0) < ɛ, n > n 0. Note first that the distance between a generic term of the sequence a n and 0 is since each a n is positive. d(a n, 0) = a n 0 = a n = a n, We need to find an n 0 N such that all terms of the subsequence {a n } n>n0 satisfy a n < ɛ, n > n 0. Since a n < a n0 for all n > n 0, this condition is satisfied if a n0 < ɛ, which is equivalent to n 0 < ɛ. Therefore, it suffices to pick any n 0 such that n 0 > ɛ to satisfy the condition d(a n, 0) < ɛ, n > n 0.
4 Week 2: Sequences and Series 2-4 Thus, we have just shown that, for any ɛ, we are able to find n 0 N such that all terms of the subsequence {a n } n>n0 have distance from zero less than ɛ. It then follows that 0 is the limit of the sequence {a n }. Example: Kepler observed that the ratio of consecutive Fibonacci numbers converges. The limit is a number known as the golden ratio ϕ: F n+ ϕ = lim. n F n It is intriguingly remarkable that the sequence {2/, 3/2, 5/3, 8/5, 3/8, 2/3, 34/2,...}, which are the ratios of Fibonacci numbers, converges to ϕ, whose value is The limit of a sequence in general. We now deal with the more general case where the terms of the sequence are not necessarily real numbers. We need a function d : A A R that associates to any couple of elements of A a real number measuring how far these two elements are. For example, if a and a are two elements of A, d(a, a ) needs to be a real number measuring the distance between a and a. 2. A function d : A A R is considered a valid distance function (and it is called a metric on A) if it satisfies some properties. 3. Let A be a set of objects. Let d : A A R. d is considered a valid distance function (in which case it is called a metric on A) if, for any a, a and a belonging to A:. Non-negativity: d(a, a ) 0; 2. Identity of indiscernibles: d(a, a ) if and only if a = a ; 3. Symmetry: d(a, a ) = d(a, a); 4. Triangle inequality: d(a, a ) + d(a, a ) d(a, a ). 4. All four properties are very intuitive. Property () says that the distance between two points cannot be a negative number. Property (2) says that the distance between two points is zero if and only if the two points coincide. Property (3) says that the distance from a to a is the same as the distance from a to a.
5 Week 2: Sequences and Series 2-5 Property (4) says that the distance one covers when one goes from a to a directly is less than (or equal to) the distance one covers when one goes from a to a passing through a third point a. In other words, if a is not on the way from a to a, the distance to be covered increases. 5. Whenever we are faced with a sequence of objects and we want to assess whether it is convergent, we need to first define a distance function on the set of objects and verify that the proposed distance function satisfies all the properties of a proper distance function. 6. Having defined the concept of a metric, we are now ready to state the formal definition of a limit of a sequence. Definition: Let A be a set of objects. Let d : A A R be a metric on A. We say that a A is a limit of a sequence {a n } of objects belonging to A, if: ɛ > 0, n 0 N : d(a n, a) < ɛ, n > n 0. If a is a limit of the sequence {a n }, we say that the sequence is a convergent sequence and that it converges to a, and write a = lim n a n. 7. A convergent sequence has a unique limit, i.e., if {a n } has a limit a, then a is the only limit of {a n }. Proof: (by contradiction) Suppose that a and a are two limits of a sequence {a n } and a a. By combining property ) and property 2) of a metric, it must be that d(a, a ) = d > 0. Pick any term a n of the sequence. By property 4) of a metric (the triangle inequality), we have d(a, a n ) + d(a n, a ) d(a, a ). It follows that d(a, a n ) + d(a n, a ) d > 0. Now, take any ɛ < d. Since a is a limit of the sequence, we can find n 0 such that d(a, a n ) < ɛ, n > n 0, which means that ɛ + d(a n, a ) d(a, a n ) + d(a n, a ) d > 0, n > n 0, and d(a n, a ) d ɛ > 0, n > n 0.
6 Week 2: Sequences and Series 2-6 Therefore, d(a n, a ) cannot be made smaller than d ɛ and consequently, a cannot be a limit of the sequence. 8. In practice, it is usually difficult to assess the convergence of a sequence using the above definition. Instead, convergence can be assessed using the following criterion: Lemma (criterion for convergence) Let A be a set of objects. Let d : A A R be a metric on A. Let {a n } be a sequence of objects belonging to A and a A. Then {a n } converges to a if and only if Proof: lim d(a n, a) = 0. n This is easily proved by defining a sequence of real numbers {d n } whose generic term is d n = d(a n, a). Note that the definition of convergence of {a n } to a, which is ɛ > 0, n 0 N : d(a n, a) < ɛ, n > n 0. Equivalently, ɛ > 0, n 0 N : d n 0 < ɛ, n > n 0. This is none other than the definition of the convergence of {d n } to So, in practice, the problem of assessing the convergence of a generic sequence of objects is simplified as follows:. Find a metric d(a n, a) to measure the distance between the terms of the sequence a n and the candidate limit a; 2. Define a new sequence {d n }, where d n = d(a n, a); 3 Study the convergence of the sequence {d n }, which is a simpler problem because {d n } is a sequence of real numbers. 2.5 Convergence of a monotone sequence of real numbers. A sequence {a n } of numbers is said to be monotone when a n a n+ for all n.
7 Week 2: Sequences and Series If {a n } is a monotone sequence of real numbers, then this sequence has a finite limit if and only if the sequence is bounded. We shall prove that if an increasing sequence {a n } is bounded above, then it is convergent and the limit is sup{a n }. n Since {a n } is non-empty and by assumption, it is bounded above, then, by the least upper bound property of real numbers, c = sup n {a n } exists and is finite. Now for every ε > 0, there exists a n0 such that a n0 > c ε, since otherwise c ε is an upper bound of {a n }, which contradicts to c being sup n {a n }. Then since {a n } is increasing, n > n 0, c a n = c a n c a N < ε, hence by definition, the limit of {a n } is sup n {a n }. 2.6 The Bolzano-Weierstrass theorem. The Bolzano-Weierstrass theorem is a fundamental result about convergence in R. The theorem states that each bounded sequence in R has a convergent subsequence. 2. Lemma: Every sequence {x n } in R has a monotone subsequence. Proof: Let us call a positive integer n a peak of the sequence if m > n implies x n > x m i.e., if x n is greater than every subsequent term in the sequence. Suppose first that the sequence has infinitely many peaks, n < n 2 < < n j < Then the subsequence {x nj } corresponding to peaks is monotonically decreasing, and a monotone subsequence is found. Suppose next that there are only finitely many peaks, let N be the last peak and n = N +. Then n is not a peak, since n > N, which implies the existence of an n 2 > n with x n2 x n. Again, n 2 > N is not a peak, hence there is n 3 > n 2 with x n3 x n2. Repeating this process leads to an infinite non-decreasing subsequence x n x n2 x n3. In this way, a monotone subsequence { } x ni is constructed. 3. The proof of Bolzano-Weierstrass theorem is to apply this lemma. Suppose we have a bounded sequence in R; by the Lemma there exists a monotone subsequence, necessarily bounded. It follows from the earlier result (a monotone and bounded sequence of real numbers has a finite limit) this subsequence of real numbers, being monotone and bounded, must converge.
8 Week 2: Sequences and Series Bounded theorem. The boundedness theorem states that a continuous function f in the closed interval [a, b] is bounded on that interval. That is, there exist real numbers m and M such that: m f(x) M for all x [a, b]. Proof: (by contradiction) Suppose the function f is not bounded above on the interval [a, b]. Then, by the property of the real numbers, for every natural number n, there exists an x n [a, b] such that f(x n ) > n. This defines a sequence {x n }. Because [a, b] is bounded, the Bolzano-Weierstrass theorem implies that there exists a convergent subsequence {x nk } of {x n }. Denote its limit by x. As [a, b] is closed, it contains x. { Because f is continuous at x, we know that f ( ) } x nk converges to the real number f(x) (as f is sequentially continuous at x.) On the other hand, f ( x nk ) > nk k for every k, which implies that +, a contradiction. Therefore, f must be bounded above on [a, b]. The proof that f attains its lower bound is similar. { f ( x nk ) } diverges to 2.8 Extreme value theorem. The extreme value theorem states that if a real-valued function f is continuous in the closed and bounded interval [a, b], then f must attain its maximum and minimum value, each at least once. That is, there exist numbers c and d in [a, b] such that f(c) f(x) f(d) for all x [a, b]. Proof: We look at the proof for the upper bound and the maximum of f. By applying these results to the function f, the existence of the lower bound and the result for the minimum of f follows. Also note that everything in the proof is done within the context of the real numbers. By the boundedness theorem, f is bounded from above, and by the least-upper-bound property of real numbers, the least upper bound (supremum) M of f exists. It is necessary to find a d in [a, b] such that M = f(d). Let n be a natural number. As M is the least upper bound, M /n is not an upper bound for f. Therefore, there exists d n in [a, b] so that M /n < f ( ) d n. This defines a sequence {dn }.
9 Week 2: Sequences and Series 2-9 Since M is an upper bound for f, we have M /n < f(d n ) M for all n. Therefore, the sequence { f(d n ) } converges to M. The Bolzano-Weierstrass theorem tells us that there exists a subsequence { d nk }, which converges to some d and, as [a, b] is closed, d is in [a, b]. Since f is continuous at d, the sequence { f(dnk ) } converges to f(d). But { f ( d nk ) } is a subsequence of {f(d n )} that converges to M, so M = f(d). Therefore, f attains its supremum M at d. 2.9 What is a series?. A series is an infinite ordered set of terms combined together by the addition operator. The term infinite series is sometimes used to emphasize the fact that series contain an infinite number of terms. 2. The order of the terms in a series can matter. By a suitable rearrangement of terms, a so-called conditionally convergent series may be made to converge to any desired value, or to diverge. 3. If the difference between successive terms of a series is a constant, then the series is said to be an arithmetic series. A series for which the ratio of each two consecutive terms a k+ /a k is a constant function of the summation index k is called a geometric series. The more general case of the ratio is a rational function of k produces a series called a hypergeometric series. 4. A series may converge to a definite value, or may not, in which case it is called divergent. Let the terms in a series be denoted a i, let the k-th partial sum be given by S k = k a i, i= and let the sequence of partial sums be given by {S = a, S 2 = a + a 2, S 3 = a + a 2 + a 3,...}. If the sequence of partial sums converges to a definite value, the series is said to converge. On the other hand, if the sequence of partial sums does not converge to a limit (e.g., it oscillates or approaches ± ), the series is said to diverge. Example: The Maclaurin series for a geometric series. Namely, for any number x whose absolute value x < is x x = x n = + x + x 2 +. n=0
10 Week 2: Sequences and Series 2-0 Example: An example of a convergent series is the geometric series. n=0 ( ) n = 2. 2 This is a special case of the Maclaurin series with x = /2. Example: An example of a divergent series is the harmonic series. n =. Example: Interestingly, while the harmonic series diverges to infinity, the alternating harmonic series converges to the natural logarithm of 2, ( ) n n = ln 2. Example: The series representation of e is e = where n! is the factorial, which is a compact symbol to denote n (n ) 2. Example: Show that the series n 2 converges. n=0 n!. S n = n 2 < n(n ) ( = + ) ( ) ( n ) n = 2 n < 2. Since the series is bounded by 2 for any n, it is necessarily convergent. In fact, Euler showed that n 2 = π A series of terms a n is said to be absolutely convergent if the series formed by taking the absolute values of the a n, i.e., n a n, converges. 6. An especially strong type of convergence is called uniform convergence, and series which are uniformly convergent have particularly nice properties. For example, the sum of a uniformly
11 Week 2: Sequences and Series 2- convergent series of continuous functions is continuous. A convergent series can be differentiated term by term, provided that the functions of the series have continuous derivatives and that the series of derivatives is uniformly convergent. Finally, a uniformly convergent series of continuous functions can be integrated term by term. 2.0 Exercises A. Suppose a n is a divergent series where each a n is strictly positive. Prove the following (a) The series (b) But the series B. Prove that the series a n + a n is also divergent. a n + n 2 a n converges. ( ) n converges. n C. Prove that if a decreasing sequence {a n } is bounded below, then it is convergent and the limit is inf n {a n}. D. Complete the extreme value theorem for the infimum m.
Problem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationWe are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero
Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.
More informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall.
.1 Limits of Sequences. CHAPTER.1.0. a) True. If converges, then there is an M > 0 such that M. Choose by Archimedes an N N such that N > M/ε. Then n N implies /n M/n M/N < ε. b) False. = n does not converge,
More informationAnalysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered
More informationChapter One. The Real Number System
Chapter One. The Real Number System We shall give a quick introduction to the real number system. It is imperative that we know how the set of real numbers behaves in the way that its completeness and
More informationSequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.
Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence
More informationEcon Slides from Lecture 1
Econ 205 Sobel Econ 205 - Slides from Lecture 1 Joel Sobel August 23, 2010 Warning I can t start without assuming that something is common knowledge. You can find basic definitions of Sets and Set Operations
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationDefinition 2.1. A metric (or distance function) defined on a non-empty set X is a function d: X X R that satisfies: For all x, y, and z in X :
MATH 337 Metric Spaces Dr. Neal, WKU Let X be a non-empty set. The elements of X shall be called points. We shall define the general means of determining the distance between two points. Throughout we
More informationMA103 Introduction to Abstract Mathematics Second part, Analysis and Algebra
206/7 MA03 Introduction to Abstract Mathematics Second part, Analysis and Algebra Amol Sasane Revised by Jozef Skokan, Konrad Swanepoel, and Graham Brightwell Copyright c London School of Economics 206
More informationThe Real Number System
MATH 337 The Real Number System Sets of Numbers Dr. Neal, WKU A set S is a well-defined collection of objects, with well-defined meaning that there is a specific description from which we can tell precisely
More informationLimits and Continuity
Chapter Limits and Continuity. Limits of Sequences.. The Concept of Limit and Its Properties A sequence { } is an ordered infinite list x,x,...,,... The n-th term of the sequence is, and n is the index
More informationMATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE
MATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE SEBASTIEN VASEY These notes describe the material for November 26, 2018 (while similar content is in Abbott s book, the presentation here is different).
More informationCopyright c 2007 Jason Underdown Some rights reserved. statement. sentential connectives. negation. conjunction. disjunction
Copyright & License Copyright c 2007 Jason Underdown Some rights reserved. statement sentential connectives negation conjunction disjunction implication or conditional antecedant & consequent hypothesis
More informationTheorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers
Page 1 Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property Suppose is an ordered set with the least-upper-bound property Suppose, and is bounded below be the set of lower
More informationNotes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.
Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3
More informationA LITTLE REAL ANALYSIS AND TOPOLOGY
A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set
More informationIntroduction to Real Analysis
Introduction to Real Analysis Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 1 Sets Sets are the basic objects of mathematics. In fact, they are so basic that
More information2.2 Some Consequences of the Completeness Axiom
60 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.2 Some Consequences of the Completeness Axiom In this section, we use the fact that R is complete to establish some important results. First, we will prove that
More informationWe have been going places in the car of calculus for years, but this analysis course is about how the car actually works.
Analysis I We have been going places in the car of calculus for years, but this analysis course is about how the car actually works. Copier s Message These notes may contain errors. In fact, they almost
More informationContents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3
Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability
More informationWalker Ray Econ 204 Problem Set 3 Suggested Solutions August 6, 2015
Problem 1. Take any mapping f from a metric space X into a metric space Y. Prove that f is continuous if and only if f(a) f(a). (Hint: use the closed set characterization of continuity). I make use of
More informationMATH 102 INTRODUCTION TO MATHEMATICAL ANALYSIS. 1. Some Fundamentals
MATH 02 INTRODUCTION TO MATHEMATICAL ANALYSIS Properties of Real Numbers Some Fundamentals The whole course will be based entirely on the study of sequence of numbers and functions defined on the real
More information1. Theorem. (Archimedean Property) Let x be any real number. There exists a positive integer n greater than x.
Advanced Calculus I, Dr. Block, Chapter 2 notes. Theorem. (Archimedean Property) Let x be any real number. There exists a positive integer n greater than x. 2. Definition. A sequence is a real-valued function
More informationChapter 3 Continuous Functions
Continuity is a very important concept in analysis. The tool that we shall use to study continuity will be sequences. There are important results concerning the subsets of the real numbers and the continuity
More informationLogical Connectives and Quantifiers
Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then
More informationLecture 3. Econ August 12
Lecture 3 Econ 2001 2015 August 12 Lecture 3 Outline 1 Metric and Metric Spaces 2 Norm and Normed Spaces 3 Sequences and Subsequences 4 Convergence 5 Monotone and Bounded Sequences Announcements: - Friday
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationLecture 5 - Hausdorff and Gromov-Hausdorff Distance
Lecture 5 - Hausdorff and Gromov-Hausdorff Distance August 1, 2011 1 Definition and Basic Properties Given a metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. If A is
More informationMath 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1
Math 8B Solutions Charles Martin March 6, Homework Problems. Let (X i, d i ), i n, be finitely many metric spaces. Construct a metric on the product space X = X X n. Proof. Denote points in X as x = (x,
More information2.1 Convergence of Sequences
Chapter 2 Sequences 2. Convergence of Sequences A sequence is a function f : N R. We write f) = a, f2) = a 2, and in general fn) = a n. We usually identify the sequence with the range of f, which is written
More informationChapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
More informationECARES Université Libre de Bruxelles MATH CAMP Basic Topology
ECARES Université Libre de Bruxelles MATH CAMP 03 Basic Topology Marjorie Gassner Contents: - Subsets, Cartesian products, de Morgan laws - Ordered sets, bounds, supremum, infimum - Functions, image, preimage,
More informationSequences. Limits of Sequences. Definition. A real-valued sequence s is any function s : N R.
Sequences Limits of Sequences. Definition. A real-valued sequence s is any function s : N R. Usually, instead of using the notation s(n), we write s n for the value of this function calculated at n. We
More informationLimit and Continuity
Limit and Continuity Table of contents. Limit of Sequences............................................ 2.. Definitions and properties...................................... 2... Definitions............................................
More informationSolutions to Tutorial 8 (Week 9)
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/
More informationConsequences of the Completeness Property
Consequences of the Completeness Property Philippe B. Laval KSU Today Philippe B. Laval (KSU) Consequences of the Completeness Property Today 1 / 10 Introduction In this section, we use the fact that R
More informationPOL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005
POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1
More informationIntroduction to Real Analysis
Christopher Heil Introduction to Real Analysis Chapter 0 Online Expanded Chapter on Notation and Preliminaries Last Updated: January 9, 2018 c 2018 by Christopher Heil Chapter 0 Notation and Preliminaries:
More informationUNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE. B.Sc. MATHEMATICS V SEMESTER. (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE B.Sc. MATHEMATICS V SEMESTER (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS QUESTION BANK 1. Find the number of elements in the power
More informationMATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.
MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if
More informationFoundations of Mathematical Analysis
Foundations of Mathematical Analysis Fabio Bagagiolo Dipartimento di Matematica, Università di Trento email:fabio.bagagiolo@unitn.it Contents 1 Introduction 2 2 Basic concepts in mathematical analysis
More informationREAL VARIABLES: PROBLEM SET 1. = x limsup E k
REAL VARIABLES: PROBLEM SET 1 BEN ELDER 1. Problem 1.1a First let s prove that limsup E k consists of those points which belong to infinitely many E k. From equation 1.1: limsup E k = E k For limsup E
More informationContinuity. Matt Rosenzweig
Continuity Matt Rosenzweig Contents 1 Continuity 1 1.1 Rudin Chapter 4 Exercises........................................ 1 1.1.1 Exercise 1............................................. 1 1.1.2 Exercise
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationReal Analysis - Notes and After Notes Fall 2008
Real Analysis - Notes and After Notes Fall 2008 October 29, 2008 1 Introduction into proof August 20, 2008 First we will go through some simple proofs to learn how one writes a rigorous proof. Let start
More informationSome Background Material
Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important
More information1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty.
1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. Let E be a subset of R. We say that E is bounded above if there exists a real number U such that x U for
More informationQF101: Quantitative Finance August 22, Week 1: Functions. Facilitator: Christopher Ting AY 2017/2018
QF101: Quantitative Finance August 22, 2017 Week 1: Functions Facilitator: Christopher Ting AY 2017/2018 The chief function of the body is to carry the brain around. Thomas A. Edison 1.1 What is a function?
More informationMATH 409 Advanced Calculus I Lecture 7: Monotone sequences. The Bolzano-Weierstrass theorem.
MATH 409 Advanced Calculus I Lecture 7: Monotone sequences. The Bolzano-Weierstrass theorem. Limit of a sequence Definition. Sequence {x n } of real numbers is said to converge to a real number a if for
More informationFoundations of Mathematical Analysis
Foundations of Mathematical Analysis Fabio Bagagiolo Dipartimento di Matematica, Università di Trento email:bagagiol@science.unitn.it Contents 1 Introduction 3 2 Basic concepts in mathematical analysis
More informationWe are now going to go back to the concept of sequences, and look at some properties of sequences in R
4 Lecture 4 4. Real Sequences We are now going to go back to the concept of sequences, and look at some properties of sequences in R Definition 3 A real sequence is increasing if + for all, and strictly
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationPart IA Numbers and Sets
Part IA Numbers and Sets Definitions Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationUndergraduate 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
More informationMA651 Topology. Lecture 9. Compactness 2.
MA651 Topology. Lecture 9. Compactness 2. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology
More informationThe Lebesgue Integral
The Lebesgue Integral Brent Nelson In these notes we give an introduction to the Lebesgue integral, assuming only a knowledge of metric spaces and the iemann integral. For more details see [1, Chapters
More informationSolutions Manual for Homework Sets Math 401. Dr Vignon S. Oussa
1 Solutions Manual for Homework Sets Math 401 Dr Vignon S. Oussa Solutions Homework Set 0 Math 401 Fall 2015 1. (Direct Proof) Assume that x and y are odd integers. Then there exist integers u and v such
More informationMidterm Review Math 311, Spring 2016
Midterm Review Math 3, Spring 206 Material Review Preliminaries and Chapter Chapter 2. Set theory (DeMorgan s laws, infinite collections of sets, nested sets, cardinality) 2. Functions (image, preimage,
More informationMAS221 Analysis, Semester 1,
MAS221 Analysis, Semester 1, 2018-19 Sarah Whitehouse Contents About these notes 2 1 Numbers, inequalities, bounds and completeness 2 1.1 What is analysis?.......................... 2 1.2 Irrational numbers.........................
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationABSTRACT INTEGRATION CHAPTER ONE
CHAPTER ONE ABSTRACT INTEGRATION Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Suggestions and errors are invited and can be mailed
More informationM17 MAT25-21 HOMEWORK 6
M17 MAT25-21 HOMEWORK 6 DUE 10:00AM WEDNESDAY SEPTEMBER 13TH 1. To Hand In Double Series. The exercises in this section will guide you to complete the proof of the following theorem: Theorem 1: Absolute
More informationAbsolute Convergence in Ordered Fields
Absolute Convergence in Ordered Fields Kristine Hampton Abstract This paper reviews Absolute Convergence in Ordered Fields by Clark and Diepeveen [1]. Contents 1 Introduction 2 2 Definition of Terms 2
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need
More informationIntroductory Analysis I Fall 2014 Homework #9 Due: Wednesday, November 19
Introductory Analysis I Fall 204 Homework #9 Due: Wednesday, November 9 Here is an easy one, to serve as warmup Assume M is a compact metric space and N is a metric space Assume that f n : M N for each
More informationProblem set 1, Real Analysis I, Spring, 2015.
Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n
More informationEC 521 MATHEMATICAL METHODS FOR ECONOMICS. Lecture 1: Preliminaries
EC 521 MATHEMATICAL METHODS FOR ECONOMICS Lecture 1: Preliminaries Murat YILMAZ Boğaziçi University In this lecture we provide some basic facts from both Linear Algebra and Real Analysis, which are going
More informationMath 117: Infinite Sequences
Math 7: Infinite Sequences John Douglas Moore November, 008 The three main theorems in the theory of infinite sequences are the Monotone Convergence Theorem, the Cauchy Sequence Theorem and the Subsequence
More informationG1CMIN Measure and Integration
G1CMIN Measure and Integration 2003-4 Prof. J.K. Langley May 13, 2004 1 Introduction Books: W. Rudin, Real and Complex Analysis ; H.L. Royden, Real Analysis (QA331). Lecturer: Prof. J.K. Langley (jkl@maths,
More information1. Let A R be a nonempty set that is bounded from above, and let a be the least upper bound of A. Show that there exists a sequence {a n } n N
Applied Analysis prelim July 15, 216, with solutions Solve 4 of the problems 1-5 and 2 of the problems 6-8. We will only grade the first 4 problems attempted from1-5 and the first 2 attempted from problems
More informationON THE POSSIBLE QUANTITIES OF FIBONACCI NUMBERS THAT OCCUR IN SOME TYPES OF INTERVALS
Acta Math. Univ. Comenianae Vol. LXXXVII, 2 (2018), pp. 291 299 291 ON THE POSSIBLE QUANTITIES OF FIBONACCI NUMBERS THAT OCCUR IN SOME TYPES OF INTERVALS B. FARHI Abstract. In this paper, we show that
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More informationMATH 131A: REAL ANALYSIS (BIG IDEAS)
MATH 131A: REAL ANALYSIS (BIG IDEAS) Theorem 1 (The Triangle Inequality). For all x, y R we have x + y x + y. Proposition 2 (The Archimedean property). For each x R there exists an n N such that n > x.
More informationAnalysis I. Classroom Notes. H.-D. Alber
Analysis I Classroom Notes H-D Alber Contents 1 Fundamental notions 1 11 Sets 1 12 Product sets, relations 5 13 Composition of statements 7 14 Quantifiers, negation of statements 9 2 Real numbers 11 21
More informationNumerical Sequences and Series
Numerical Sequences and Series Written by Men-Gen Tsai email: b89902089@ntu.edu.tw. Prove that the convergence of {s n } implies convergence of { s n }. Is the converse true? Solution: Since {s n } is
More information6.2 Deeper Properties of Continuous Functions
6.2. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 69 6.2 Deeper Properties of Continuous Functions 6.2. Intermediate Value Theorem and Consequences When one studies a function, one is usually interested in
More information5.5 Deeper Properties of Continuous Functions
5.5. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 195 5.5 Deeper Properties of Continuous Functions 5.5.1 Intermediate Value Theorem and Consequences When one studies a function, one is usually interested
More informationThe Two Faces of Infinity Dr. Bob Gardner Great Ideas in Science (BIOL 3018)
The Two Faces of Infinity Dr. Bob Gardner Great Ideas in Science (BIOL 3018) From the webpage of Timithy Kohl, Boston University INTRODUCTION Note. We will consider infinity from two different perspectives:
More informationPart 2 Continuous functions and their properties
Part 2 Continuous functions and their properties 2.1 Definition Definition A function f is continuous at a R if, and only if, that is lim f (x) = f (a), x a ε > 0, δ > 0, x, x a < δ f (x) f (a) < ε. Notice
More information5 Set Operations, Functions, and Counting
5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,
More informationPrinciple of Mathematical Induction
Advanced Calculus I. Math 451, Fall 2016, Prof. Vershynin Principle of Mathematical Induction 1. Prove that 1 + 2 + + n = 1 n(n + 1) for all n N. 2 2. Prove that 1 2 + 2 2 + + n 2 = 1 n(n + 1)(2n + 1)
More informationSequences. We know that the functions can be defined on any subsets of R. As the set of positive integers
Sequences We know that the functions can be defined on any subsets of R. As the set of positive integers Z + is a subset of R, we can define a function on it in the following manner. f: Z + R f(n) = a
More informationANALYSIS Lecture Notes
MA2730 ANALYSIS Lecture Notes Martins Bruveris 206 Contents Sequences 5. Sequences and convergence 5.2 Bounded and unbounded sequences 8.3 Properties of convergent sequences 0.4 Sequences and functions
More informationSets, Functions and Metric Spaces
Chapter 14 Sets, Functions and Metric Spaces 14.1 Functions and sets 14.1.1 The function concept Definition 14.1 Let us consider two sets A and B whose elements may be any objects whatsoever. Suppose that
More informationTopology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:
Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework
More informationChapter 1 The Real Numbers
Chapter 1 The Real Numbers In a beginning course in calculus, the emphasis is on introducing the techniques of the subject;i.e., differentiation and integration and their applications. An advanced calculus
More informationFINAL EXAM Math 25 Temple-F06
FINAL EXAM Math 25 Temple-F06 Write solutions on the paper provided. Put your name on this exam sheet, and staple it to the front of your finished exam. Do Not Write On This Exam Sheet. Problem 1. (Short
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationMORE ON CONTINUOUS FUNCTIONS AND SETS
Chapter 6 MORE ON CONTINUOUS FUNCTIONS AND SETS This chapter can be considered enrichment material containing also several more advanced topics and may be skipped in its entirety. You can proceed directly
More informationAdvanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010
Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with
More informationEconomics 204 Summer/Fall 2011 Lecture 2 Tuesday July 26, 2011 N Now, on the main diagonal, change all the 0s to 1s and vice versa:
Economics 04 Summer/Fall 011 Lecture Tuesday July 6, 011 Section 1.4. Cardinality (cont.) Theorem 1 (Cantor) N, the set of all subsets of N, is not countable. Proof: Suppose N is countable. Then there
More informationDefinitions & Theorems
Definitions & Theorems Math 147, Fall 2009 December 19, 2010 Contents 1 Logic 2 1.1 Sets.................................................. 2 1.2 The Peano axioms..........................................
More informationProof. We indicate by α, β (finite or not) the end-points of I and call
C.6 Continuous functions Pag. 111 Proof of Corollary 4.25 Corollary 4.25 Let f be continuous on the interval I and suppose it admits non-zero its (finite or infinite) that are different in sign for x tending
More informationCHAPTER I THE RIESZ REPRESENTATION THEOREM
CHAPTER I THE RIESZ REPRESENTATION THEOREM We begin our study by identifying certain special kinds of linear functionals on certain special vector spaces of functions. We describe these linear functionals
More informationmeans is a subset of. So we say A B for sets A and B if x A we have x B holds. BY CONTRAST, a S means that a is a member of S.
1 Notation For those unfamiliar, we have := means equal by definition, N := {0, 1,... } or {1, 2,... } depending on context. (i.e. N is the set or collection of counting numbers.) In addition, means for
More informationBootcamp. Christoph Thiele. Summer As in the case of separability we have the following two observations: Lemma 1 Finite sets are compact.
Bootcamp Christoph Thiele Summer 212.1 Compactness Definition 1 A metric space is called compact, if every cover of the space has a finite subcover. As in the case of separability we have the following
More informationFrom now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1.
Chapter 1 Metric spaces 1.1 Metric and convergence We will begin with some basic concepts. Definition 1.1. (Metric space) Metric space is a set X, with a metric satisfying: 1. d(x, y) 0, d(x, y) = 0 x
More information