MAS221 Analysis Semester Chapter 2 problems

Size: px
Start display at page:

Download "MAS221 Analysis Semester Chapter 2 problems"

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

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 information

Principle of Mathematical Induction

Principle 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 information

MA103 Introduction to Abstract Mathematics Second part, Analysis and Algebra

MA103 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 information

Scalar multiplication and addition of sequences 9

Scalar 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 information

C.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 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 information

Numerical Sequences and Series

Numerical 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 information

Sequences. 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. 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 information

Chapter 5. Measurable Functions

Chapter 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 information

Chapter 2. Real Numbers. 1. Rational Numbers

Chapter 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 information

Part 2 Continuous functions and their properties

Part 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 information

Sequences. 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 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 information

Set, functions and Euclidean space. Seungjin Han

Set, 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 information

MATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE

MATH 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 information

Supremum and Infimum

Supremum 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 information

211 Real Analysis. f (x) = x2 1. x 1. x 2 1

211 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 information

Characterisation of Accumulation Points. Convergence in Metric Spaces. Characterisation of Closed Sets. Characterisation of Closed Sets

Characterisation 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 information

MATH 301 INTRO TO ANALYSIS FALL 2016

MATH 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 information

MATH 117 LECTURE NOTES

MATH 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 information

means 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.

means 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 information

First In-Class Exam Solutions Math 410, Professor David Levermore Monday, 1 October 2018

First 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 information

3 Measurable Functions

3 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 information

THE RADIUS OF CONVERGENCE FORMULA. a n (z c) n, f(z) =

THE 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 information

INTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES

INTRODUCTION 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 information

Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright 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 information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

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

More information

Lecture 4 Lebesgue spaces and inequalities

Lecture 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 information

Mathematics 242 Principles of Analysis Solutions for Problem Set 5 Due: March 15, 2013

Mathematics 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 information

4130 HOMEWORK 4. , a 2

4130 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 information

Econ Lecture 3. Outline. 1. Metric Spaces and Normed Spaces 2. Convergence of Sequences in Metric Spaces 3. Sequences in R and R n

Econ 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 information

Limit and Continuity

Limit and Continuity Limit and Continuity Table of contents. Limit of Sequences............................................ 2.. Definitions and properties...................................... 2... Definitions............................................

More information

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011

Economics 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 information

Sequences CHAPTER 3. Definition. A sequence is a function f : N R.

Sequences 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 information

Math 117: Infinite Sequences

Math 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 information

Econ Slides from Lecture 1

Econ 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 information

1. 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. 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 information

Chapter 1 The Real Numbers

Chapter 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 information

Midterm Review Math 311, Spring 2016

Midterm 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 information

Week 2: Sequences and Series

Week 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 information

In 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.

In 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 information

2.2 Some Consequences of the Completeness Axiom

2.2 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 information

Math 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 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 information

INFINITE SEQUENCES AND SERIES

INFINITE 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 information

From Calculus II: An infinite series is an expression of the form

From 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 information

An Analysis Sketchbook

An 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 information

Calculus (Real Analysis I)

Calculus (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 information

FINAL EXAM Math 25 Temple-F06

FINAL 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 information

Construction of a general measure structure

Construction 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 information

Chapter One. The Real Number System

Chapter 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 information

Read carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.

Read 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 information

Statistical inference

Statistical 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 information

2. Introduction to commutative rings (continued)

2. 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 information

Estimates for probabilities of independent events and infinite series

Estimates 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 information

Seunghee Ye Ma 8: Week 2 Oct 6

Seunghee 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 information

We begin by considering the following three sequences:

We 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 information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 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 information

CHAPTER 4. Series. 1. What is a Series?

CHAPTER 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 information

ANALYSIS Lecture Notes

ANALYSIS 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 information

a n b n ) n N is convergent with b n is convergent.

a 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 information

2.4 The Extreme Value Theorem and Some of its Consequences

2.4 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 information

MA131 - Analysis 1. Workbook 6 Completeness II

MA131 - 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 information

We 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

We 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 information

MATH 1231 MATHEMATICS 1B CALCULUS. Section 4: - Convergence of Series.

MATH 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 information

Real Analysis - Notes and After Notes Fall 2008

Real 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 information

Homework 1 Solutions

Homework 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 information

Lebesgue Integration: A non-rigorous introduction. What is wrong with Riemann integration?

Lebesgue 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 information

Solutions Manual for Homework Sets Math 401. Dr Vignon S. Oussa

Solutions 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 information

Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010

Advanced 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 information

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.

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. 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 information

Introduction and Preliminaries

Introduction 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 information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The 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 information

Math LM (24543) Lectures 01

Math 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 information

Lecture 1: Overview of percolation and foundational results from probability theory 30th July, 2nd August and 6th August 2007

Lecture 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 information

i. v = 0 if and only if v 0. iii. v + w v + w. (This is the Triangle Inequality.)

i. 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 information

The Lebesgue Integral

The 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 information

MAS221 Analysis, Semester 1,

MAS221 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 information

Math Camp Day 1. Tomohiro Kusano. University of Tokyo September 4, https://sites.google.com/site/utgsemathcamp2015/

Math 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.

= 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 information

4.4 Uniform Convergence of Sequences of Functions and the Derivative

4.4 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 information

MATH41011/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 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 information

Normed and Banach spaces

Normed 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 information

F (x) = P [X x[. DF1 F is nondecreasing. DF2 F is right-continuous

F (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 information

Notes 6 : First and second moment methods

Notes 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 information

ADVANCE TOPICS IN ANALYSIS - REAL. 8 September September 2011

ADVANCE 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 information

4.3 Limit of a Sequence: Theorems

4.3 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 information

Introduction to Real Analysis

Introduction 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 information

Homework 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.

Homework 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 information

Math 163 (23) - Midterm Test 1

Math 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 information

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part 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 information

Convexity 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. 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 information

Appendix A. Sequences and series. A.1 Sequences. Definition A.1 A sequence is a function N R.

Appendix 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 information

Due date: Monday, February 6, 2017.

Due 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 information

MA 1124 Solutions 14 th May 2012

MA 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 information

17. Convergence of Random Variables

17. 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 information

Solution of the 7 th Homework

Solution 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 information

Sequences. 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.

Sequences. 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 information

M17 MAT25-21 HOMEWORK 6

M17 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 information

3 (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?

3 (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 information

2 (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?

2 (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 information

Probability and Measure

Probability 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 information

STA2112F99 ε δ Review

STA2112F99 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