MAS221 Analysis Semester Chapter 2 problems
|
|
- Tyler Perkins
- 5 years ago
- Views:
Transcription
1 MAS221 Analysis Semester Chapter 2 problems 20. Consider the sequence (a n ), with general term a n = Can you n guess the limit l of this sequence? (a) Verify that your guess is plausible by finding an N N, for each of the following given values of ε such that n > N a n l < ε. (i) ε = 0.1 (ii) ε = 0.01 (iii) ε = (iv) ε = (v) ε = (b) Give a rigorous proof that (a n ) converges to l. 21. Guess the limits of the sequences whose nth terms are as follows. (a) 1 1 n, (b) 3 n, (c) 1 n 2, (d) 1 n. Then, in each case, use the definition of convergence to prove that your guesses are correct. (So given any ε > 0, you need to find a suitable N that works.) 22. Write down a formula for the general term of a sequence (a n ) such that a 1, a 2, a 3, a 4 and a 5 are precisely 1, 2, 3, 4, 5, continuing the pattern in the obvious way. Use the definition of limit to prove that the sequence converges to Show that if (x n ) converges to x then ( x n ) converges to x. Is the converse true? If so, give a proof and if not, give a counter-example. [Hint: For the first part, use Theorem ] 24. If lim a n = 0 and 0 b n a n for all n N, prove directly from the definition of convergence that lim b n = (a) If a, b 0, show that a + b a + b. [Hint: Consider the square of both sides. You may use that the square root function is increasing.] 1
2 (b) If a, b R, deduce that a b a b. [Hint: Imitate the proof of Theorem ] (c) Prove that if the sequence (a n ) converges to l, then ( a n ) converges to l. [Hint: Use the result of (b).] 26. Use the algebra of limits to find the limits of the following sequences: ( (a) 2 1 ) ( ) (, (b) ) 2 2n + 3, (c) n n n 5n + 9, (d) n n 2 n The following were all written down in an examination in answer to the question What is the definition of a sequence (x n ) converging to a limit x? Say what is wrong, if anything, with each of them. (a) For some ε > 0 there is an N such that x n x < ε for n > N. (b) Where ε > 0, for some natural number N where n > N, x n x < ε. (c) For every positive number ε there is a term in the sequence after which all the following terms are within ε of x. (d) For any ε > 0 there is some n > N such that x n x < ε. 28. The purpose of this question is to show that the order of the words in the definition of convergence is critical. Here is a definition made up just for this question to explore this, where we change around the order. A sequence (x n ) is defined to be ridiculously-convergent to x if there exists N N such that for every ε > 0 we have x n x < ε whenever n > N. (a) Comment on the difference between ridiculous convergence and convergence (in the usual sense.) (b) Show that the sequence ( 1 ) is not ridiculously-convergent to 0. n 29. Let C > 0 be a fixed positive real number. Show that the sequence (x n ) converges to x if and only if for any ε > 0 there is a natural number N such that x n x < Cε whenever n > N. 2
3 [Comment: This is just playing with the definition of convergence, but the point is that it is very often useful to be able to do this, as in the next problem. The intuition behind it is, that if ε is thought of as a quantity that can be made as small as you like, then so is Cε.] 30. A sequence (a n ) is said to be null if it converges to zero. Prove that if (a n ) is null, and (b n ) is bounded (but not necessarily convergent), then the sequence (a n b n ) is null. [Hint: Use the definition of convergence and Problem 29.] 31. (a) Show that if (x n ) is a sequence converging to l where each x n 0, then l 0. (Hint: Try a proof by contradiction.) (b) Deduce that if (x n ) is a sequence converging to x such that x n < a for all n N, then x a. Is it true that x < a? For the last assertion, give a proof if it is true, or a counter example if it is false. 32. Consider a positive sequence (x n ), i.e. one for which each x n > 0, and assume that the sequence converges to a positive limit. Show that x n+1 lim = 1. Give examples, one in each case, of a convergent x n positive sequence (x n ) for which the sequence whose nth term is x n+1 x n (i) converges to zero, (ii) converges to a half, (iii) diverges (trickier). 33. (a) Let r > 1 and consider the sequence (r 1 n ). Prove that it converges to 1. [Hint: Write r 1 n = 1 + c n where c n > 0 and use Bernoulli s inequality from Problem 5 to show that lim c n = 0.] (b) Show that lim r 1 n = 1 when 0 < r < 1. [Hint: Write r = 1 s.] (c) Prove that lim n 1 n = 1. [Hint: Write n 1 n = 1 + c n and first 1 show that lim n n = 1. ] 34. Let x 1 = 2.5 and x n+1 = 1 5 (x2 n + 6) for n > 1. (a) Show that each 2 x n 3. (Hint: Try a proof by contradiction.) (b) Show that x n+1 x n = 1 5 (x n 2)(x n 3). (c) Show that the sequence (x n ) is monotone and find its limit as n. 35. Prove that if a sequence (a n ) is monotonic decreasing, and bounded below, then it converges to its infimum. In other words, prove Theorem 3
4 2.3.2(2) [Hint: One way of doing this is to imitate the proof of Theorem 2.3.2(1) given in the lectures.] 36. Let a b > 0. We define sequences (a n ) and (b n ) by taking a 1 and b 1 to be a and b respectively, and requiring that for n 1, a n+1 = 1 2 (a n + b n ) and b n+1 = a n b n. In other words, a n+1 is the arithmetic mean of a n and b n while b n+1 is their geometric mean. (a) Prove that b n b n+1 a n+1 a n for each n. (b) Prove that a n+1 b n (a n b n ) for all n. (c) Deduce that the sequences (a n ) and (b n ) are each convergent and that they converge to the same limit. (The common limit M(a; b) = lim n a n = lim b n is called the arithmetic-geometric mean of a and b. It can be given a precise form using objects called elliptic integrals.) 37. Show that if (a n ) is a sequence that is both monotonic increasing and also convergent to a limit l as n, then (a n ) is bounded above and l = sup n N (a n ). What happens when (a n ) is monotonic decreasing and convergent? 38. The purpose of this question is to prove that n p x n 0 as n for any positive real number p and for any 1 < x < 1. Assume firstly that 0 < x < 1, and write a n = n p x n. (a) Show that lim a n+1 a n = x. (b) Deduce that a n+1 a n is eventually less than one and so (a n ) is eventually decreasing. [Here eventually means there is some N such that the statement is true for all n > N.] (c) Deduce that the sequence (a n ) tends to a non-negative limit l. (d) Use part (a) with Problem 30 to deduce that l = 0. What about the case where 1 < x < 0? 39. Suppose that (a n ) is a monotonic increasing sequence that has a subsequence (a nk ) which converges to a limit l. 4
5 (a) Show that a n l for all n N. [Hint: Use the result of Problem 37.] (b) Show that (a n ) converges to l as n. [Hint: Use the definition of convergence.] (c) What happens when increasing is replaced by decreasing in this question? 40. Let (x n ) be a bounded sequence and define two associated sequences as follows a n = sup{x m m n} and b n = inf{x m m n} (a) Show that (a n ) is monotonic decreasing, bounded below and hence convergent. (b) Show that (b n ) is monotonic increasing and bounded above and hence convergent. We define lim sup x n = lim a n, lim inf x n = lim b n. Find lim sup and lim inf of the following sequences: (i) ( 1) n, (ii) 1 n, (iii) ( 1) n (1 1 n ). Note: lim sup and lim inf play a major role in some parts of advanced analysis. An important theorem states that a bounded sequence (x n ) converges to the limit l if and only if lim sup x n = lim inf x n = l. You may encounter some books in which lim sup x n lim x n and lim inf x n is written lim x n. is written 41. Prove that every Cauchy sequence is bounded. [Hint: Imitate the proof of the fact that every convergent sequence is bounded (Theorem 2.1.8).] 5
6 42. Prove that every convergent sequence is Cauchy. [Hint: Suppose (a n ) converges to a; think about how you might show that a n a m is bounded by the sum of two terms, each smaller than ε/2, for sufficiently large m and n.] 43. Show that (0, 1] is not complete by finding an example of a Cauchy sequence, all of whose terms lie in this interval, which converges to a limit that is not in the interval. 6
2.1 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 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 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 informationScalar multiplication and addition of sequences 9
8 Sequences 1.2.7. Proposition. Every subsequence of a convergent sequence (a n ) n N converges to lim n a n. Proof. If (a nk ) k N is a subsequence of (a n ) n N, then n k k for every k. Hence if ε >
More informationC.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series
C.7 Numerical series Pag. 147 Proof of the converging criteria for series Theorem 5.29 (Comparison test) Let and be positive-term series such that 0, for any k 0. i) If the series converges, then also
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 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 informationChapter 5. Measurable Functions
Chapter 5. Measurable Functions 1. Measurable Functions Let X be a nonempty set, and let S be a σ-algebra of subsets of X. Then (X, S) is a measurable space. A subset E of X is said to be measurable if
More informationChapter 2. Real Numbers. 1. Rational Numbers
Chapter 2. Real Numbers 1. Rational Numbers A commutative ring is called a field if its nonzero elements form a group under multiplication. Let (F, +, ) be a filed with 0 as its additive identity element
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 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 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 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 informationSupremum and Infimum
Supremum and Infimum UBC M0 Lecture Notes by Philip D. Loewen The Real Number System. Work hard to construct from the axioms a set R with special elements O and I, and a subset P R, and mappings A: R R
More information211 Real Analysis. f (x) = x2 1. x 1. x 2 1
Part. Limits of functions. Introduction 2 Real Analysis Eample. What happens to f : R \ {} R, given by f () = 2,, as gets close to? If we substitute = we get f () = 0 which is undefined. Instead we 0 might
More informationCharacterisation of Accumulation Points. Convergence in Metric Spaces. Characterisation of Closed Sets. Characterisation of Closed Sets
Convergence in Metric Spaces Functional Analysis Lecture 3: Convergence and Continuity in Metric Spaces Bengt Ove Turesson September 4, 2016 Suppose that (X, d) is a metric space. A sequence (x n ) X is
More informationMATH 301 INTRO TO ANALYSIS FALL 2016
MATH 301 INTRO TO ANALYSIS FALL 016 Homework 04 Professional Problem Consider the recursive sequence defined by x 1 = 3 and +1 = 1 4 for n 1. (a) Prove that ( ) converges. (Hint: show that ( ) is decreasing
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 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 informationFirst In-Class Exam Solutions Math 410, Professor David Levermore Monday, 1 October 2018
First In-Class Exam Solutions Math 40, Professor David Levermore Monday, October 208. [0] Let {b k } k N be a sequence in R and let A be a subset of R. Write the negations of the following assertions.
More information3 Measurable Functions
3 Measurable Functions Notation A pair (X, F) where F is a σ-field of subsets of X is a measurable space. If µ is a measure on F then (X, F, µ) is a measure space. If µ(x) < then (X, F, µ) is a probability
More informationTHE RADIUS OF CONVERGENCE FORMULA. a n (z c) n, f(z) =
THE RADIUS OF CONVERGENCE FORMULA Every complex power series, f(z) = (z c) n, n=0 has a radius of convergence, nonnegative-real or infinite, R = R(f) [0, + ], that describes the convergence of the series,
More informationINTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES
INTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES You will be expected to reread and digest these typed notes after class, line by line, trying to follow why the line is true, for example how it
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 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 informationLecture 4 Lebesgue spaces and inequalities
Lecture 4: Lebesgue spaces and inequalities 1 of 10 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 4 Lebesgue spaces and inequalities Lebesgue spaces We have seen how
More informationMathematics 242 Principles of Analysis Solutions for Problem Set 5 Due: March 15, 2013
Mathematics Principles of Analysis Solutions for Problem Set 5 Due: March 15, 013 A Section 1. For each of the following sequences, determine three different subsequences, each converging to a different
More information4130 HOMEWORK 4. , a 2
4130 HOMEWORK 4 Due Tuesday March 2 (1) Let N N denote the set of all sequences of natural numbers. That is, N N = {(a 1, a 2, a 3,...) : a i N}. Show that N N = P(N). We use the Schröder-Bernstein Theorem.
More informationEcon Lecture 3. Outline. 1. Metric Spaces and Normed Spaces 2. Convergence of Sequences in Metric Spaces 3. Sequences in R and R n
Econ 204 2011 Lecture 3 Outline 1. Metric Spaces and Normed Spaces 2. Convergence of Sequences in Metric Spaces 3. Sequences in R and R n 1 Metric Spaces and Metrics Generalize distance and length notions
More informationLimit and Continuity
Limit and Continuity Table of contents. Limit of Sequences............................................ 2.. Definitions and properties...................................... 2... Definitions............................................
More informationEconomics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011
Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Section 2.6 (cont.) Properties of Real Functions Here we first study properties of functions from R to R, making use of the additional structure
More informationSequences CHAPTER 3. Definition. A sequence is a function f : N R.
CHAPTER 3 Sequences 1. Limits and the Archimedean Property Our first basic object for investigating real numbers is the sequence. Before we give the precise definition of a sequence, we will give the intuitive
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 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 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 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 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 informationWeek 2: Sequences and Series
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
More informationIn last semester, we have seen some examples about it (See Tutorial Note #13). Try to have a look on that. Here we try to show more technique.
MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #4 Part I: Cauchy Sequence Definition (Cauchy Sequence): A sequence of real number { n } is Cauchy if and only if for any ε >
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 informationMath 111, Introduction to the Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions
Math 111, Introduction to the Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions For each question, there is a model solution (showing you the level of detail I expect on the exam) and then below
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES Infinite sequences and series were introduced briefly in A Preview of Calculus in connection with Zeno s paradoxes and the decimal representation
More informationFrom Calculus II: An infinite series is an expression of the form
MATH 3333 INTERMEDIATE ANALYSIS BLECHER NOTES 75 8. Infinite series of numbers From Calculus II: An infinite series is an expression of the form = a m + a m+ + a m+2 + ( ) Let us call this expression (*).
More informationAn Analysis Sketchbook
An Analysis Sketchbook Jonathan K. Hodge Clark Wells Grand Valley State University c 2007, Jonathan K. Hodge and Clark Wells Contents 1 Calculus in Q? 1 Introduction................................ 1
More informationCalculus (Real Analysis I)
Calculus (Real Analysis I) (MAT122β) Department of Mathematics University of Ruhuna A.W.L. Pubudu Thilan Department of Mathematics University of Ruhuna Calculus (Real Analysis I)(MAT122β) 1/172 Chapter
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 informationConstruction of a general measure structure
Chapter 4 Construction of a general measure structure We turn to the development of general measure theory. The ingredients are a set describing the universe of points, a class of measurable subsets along
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 informationRead carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.
THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: January 2011 Analysis I Time Allowed: 1.5 hours Read carefully the instructions on the answer book and make sure that the particulars required are entered
More informationStatistical inference
Statistical inference Contents 1. Main definitions 2. Estimation 3. Testing L. Trapani MSc Induction - Statistical inference 1 1 Introduction: definition and preliminary theory In this chapter, we shall
More information2. Introduction to commutative rings (continued)
2. Introduction to commutative rings (continued) 2.1. New examples of commutative rings. Recall that in the first lecture we defined the notions of commutative rings and field and gave some examples of
More informationEstimates for probabilities of independent events and infinite series
Estimates for probabilities of independent events and infinite series Jürgen Grahl and Shahar evo September 9, 06 arxiv:609.0894v [math.pr] 8 Sep 06 Abstract This paper deals with finite or infinite sequences
More informationSeunghee Ye Ma 8: Week 2 Oct 6
Week 2 Summary This week, we will learn about sequences and real numbers. We first define what we mean by a sequence and discuss several properties of sequences. Then, we will talk about what it means
More informationWe begin by considering the following three sequences:
STUDENT S COMPANIONS IN BASIC MATH: THE TWELFTH The Concept of Limits for Sequences and Series In calculus, the concept of limits is of paramount importance, in view of the fact that many basic objects
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 informationCHAPTER 4. Series. 1. What is a Series?
CHAPTER 4 Series Given a sequence, in many contexts it is natural to ask about the sum of all the numbers in the sequence. If only a finite number of the are nonzero, this is trivial and not very interesting.
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 informationa n b n ) n N is convergent with b n is convergent.
32 Series Let s n be the n-th partial sum of n N and let t n be the n-th partial sum of n N. For k n we then have n n s n s k = a i a i = t n t k. i=k+1 i=k+1 Since t n n N is convergent by assumption,
More information2.4 The Extreme Value Theorem and Some of its Consequences
2.4 The Extreme Value Theorem and Some of its Consequences The Extreme Value Theorem deals with the question of when we can be sure that for a given function f, (1) the values f (x) don t get too big or
More informationMA131 - Analysis 1. Workbook 6 Completeness II
MA3 - Analysis Workbook 6 Completeness II Autumn 2004 Contents 3.7 An Interesting Sequence....................... 3.8 Consequences of Completeness - General Bounded Sequences.. 3.9 Cauchy Sequences..........................
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 informationMATH 1231 MATHEMATICS 1B CALCULUS. Section 4: - Convergence of Series.
MATH 23 MATHEMATICS B CALCULUS. Section 4: - Convergence of Series. The objective of this section is to get acquainted with the theory and application of series. By the end of this section students will
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 informationHomework 1 Solutions
MATH 171 Spring 2016 Problem 1 Homework 1 Solutions (If you find any errors, please send an e-mail to farana at stanford dot edu) Presenting your arguments in steps, using only axioms of an ordered field,
More informationLebesgue Integration: A non-rigorous introduction. What is wrong with Riemann integration?
Lebesgue Integration: A non-rigorous introduction What is wrong with Riemann integration? xample. Let f(x) = { 0 for x Q 1 for x / Q. The upper integral is 1, while the lower integral is 0. Yet, the function
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 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 informationCauchy s Theorem (rigorous) In this lecture, we will study a rigorous proof of Cauchy s Theorem. We start by considering the case of a triangle.
Cauchy s Theorem (rigorous) In this lecture, we will study a rigorous proof of Cauchy s Theorem. We start by considering the case of a triangle. Given a certain complex-valued analytic function f(z), for
More informationIntroduction and Preliminaries
Chapter 1 Introduction and Preliminaries This chapter serves two purposes. The first purpose is to prepare the readers for the more systematic development in later chapters of methods of real analysis
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More informationMath LM (24543) Lectures 01
Math 32300 LM (24543) Lectures 01 Ethan Akin Office: NAC 6/287 Phone: 650-5136 Email: ethanakin@earthlink.net Spring, 2018 Contents Introduction, Ross Chapter 1 and Appendix The Natural Numbers N and The
More informationLecture 1: Overview of percolation and foundational results from probability theory 30th July, 2nd August and 6th August 2007
CSL866: Percolation and Random Graphs IIT Delhi Arzad Kherani Scribe: Amitabha Bagchi Lecture 1: Overview of percolation and foundational results from probability theory 30th July, 2nd August and 6th August
More informationi. v = 0 if and only if v 0. iii. v + w v + w. (This is the Triangle Inequality.)
Definition 5.5.1. A (real) normed vector space is a real vector space V, equipped with a function called a norm, denoted by, provided that for all v and w in V and for all α R the real number v 0, and
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 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 informationMath Camp Day 1. Tomohiro Kusano. University of Tokyo September 4, https://sites.google.com/site/utgsemathcamp2015/
Math Camp 2015 Day 1 Tomohiro Kusano University of Tokyo September 4, 2015 https://sites.google.com/site/utgsemathcamp2015/ Outline I. Logic Basic Logic, T-F table, logical equivalence II. Set Theory Quantifiers,
More information= 1 2 x (x 1) + 1 {x} (1 {x}). [t] dt = 1 x (x 1) + O (1), [t] dt = 1 2 x2 + O (x), (where the error is not now zero when x is an integer.
Problem Sheet,. i) Draw the graphs for [] and {}. ii) Show that for α R, α+ α [t] dt = α and α+ α {t} dt =. Hint Split these integrals at the integer which must lie in any interval of length, such as [α,
More information4.4 Uniform Convergence of Sequences of Functions and the Derivative
4.4 Uniform Convergence of Sequences of Functions and the Derivative Say we have a sequence f n (x) of functions defined on some interval, [a, b]. Let s say they converge in some sense to a function f
More informationMATH41011/MATH61011: FOURIER SERIES AND LEBESGUE INTEGRATION. Extra Reading Material for Level 4 and Level 6
MATH41011/MATH61011: FOURIER SERIES AND LEBESGUE INTEGRATION Extra Reading Material for Level 4 and Level 6 Part A: Construction of Lebesgue Measure The first part the extra material consists of the construction
More informationNormed and Banach spaces
Normed and Banach spaces László Erdős Nov 11, 2006 1 Norms We recall that the norm is a function on a vectorspace V, : V R +, satisfying the following properties x + y x + y cx = c x x = 0 x = 0 We always
More informationF (x) = P [X x[. DF1 F is nondecreasing. DF2 F is right-continuous
7: /4/ TOPIC Distribution functions their inverses This section develops properties of probability distribution functions their inverses Two main topics are the so-called probability integral transformation
More informationNotes 6 : First and second moment methods
Notes 6 : First and second moment methods Math 733-734: Theory of Probability Lecturer: Sebastien Roch References: [Roc, Sections 2.1-2.3]. Recall: THM 6.1 (Markov s inequality) Let X be a non-negative
More informationADVANCE TOPICS IN ANALYSIS - REAL. 8 September September 2011
ADVANCE TOPICS IN ANALYSIS - REAL NOTES COMPILED BY KATO LA Introductions 8 September 011 15 September 011 Nested Interval Theorem: If A 1 ra 1, b 1 s, A ra, b s,, A n ra n, b n s, and A 1 Ě A Ě Ě A n
More information4.3 Limit of a Sequence: Theorems
4.3. LIMIT OF A SEQUENCE: THEOREMS 0 4.3 Limit of a Sequence: Theorems 4.3. Elementary Theorems In example 76, we used an approximation to simplify the problem a little bit. In this particular example,
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 informationHomework 4, 5, 6 Solutions. > 0, and so a n 0 = n + 1 n = ( n+1 n)( n+1+ n) 1 if n is odd 1/n if n is even diverges.
2..2(a) lim a n = 0. Homework 4, 5, 6 Solutions Proof. Let ɛ > 0. Then for n n = 2+ 2ɛ we have 2n 3 4+ ɛ 3 > ɛ > 0, so 0 < 2n 3 < ɛ, and thus a n 0 = 2n 3 < ɛ. 2..2(g) lim ( n + n) = 0. Proof. Let ɛ >
More informationMath 163 (23) - Midterm Test 1
Name: Id #: Math 63 (23) - Midterm Test Spring Quarter 208 Friday April 20, 09:30am - 0:20am Instructions: Prob. Points Score possible 26 2 4 3 0 TOTAL 50 Read each problem carefully. Write legibly. Show
More informationPart V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory
Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite
More informationConvexity in R n. The following lemma will be needed in a while. Lemma 1 Let x E, u R n. If τ I(x, u), τ 0, define. f(x + τu) f(x). τ.
Convexity in R n Let E be a convex subset of R n. A function f : E (, ] is convex iff f(tx + (1 t)y) (1 t)f(x) + tf(y) x, y E, t [0, 1]. A similar definition holds in any vector space. A topology is needed
More informationAppendix A. Sequences and series. A.1 Sequences. Definition A.1 A sequence is a function N R.
Appendix A Sequences and series This course has for prerequisite a course (or two) of calculus. The purpose of this appendix is to review basic definitions and facts concerning sequences and series, which
More informationDue date: Monday, February 6, 2017.
Modern Analysis Homework 3 Solutions Due date: Monday, February 6, 2017. 1. If A R define A = {x R : x A}. Let A be a nonempty set of real numbers, assume A is bounded above. Prove that A is bounded below
More informationMA 1124 Solutions 14 th May 2012
MA 1124 Solutions 14 th May 2012 1 (a) Use True/False Tables to prove (i) P = Q Q = P The definition of P = Q is given by P Q P = Q T T T T F F F T T F F T So Q P Q = P F F T T F F F T T T T T Since the
More information17. Convergence of Random Variables
7. Convergence of Random Variables In elementary mathematics courses (such as Calculus) one speaks of the convergence of functions: f n : R R, then lim f n = f if lim f n (x) = f(x) for all x in R. This
More informationSolution of the 7 th Homework
Solution of the 7 th Homework Sangchul Lee December 3, 2014 1 Preliminary In this section we deal with some facts that are relevant to our problems but can be coped with only previous materials. 1.1 Maximum
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 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 information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationProbability and Measure
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability
More informationSTA2112F99 ε δ Review
STA2112F99 ε δ Review 1. Sequences of real numbers Definition: Let a 1, a 2,... be a sequence of real numbers. We will write a n a, or lim a n = a, if for n all ε > 0, there exists a real number N such
More information