Scalar multiplication and addition of sequences 9
|
|
- Simon Lawrence
- 5 years ago
- Views:
Transcription
1 8 Sequences 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 > 0 and N N with a n r < ε for all n N, then also a nk r < ε for all k N. Notice that does not only say that every subsequence of a convergent sequence converges. The statement is that all subsequences of a convergent sequence (a n ) n N converge, and they all converge to the same number, namely to lim n a n. The assertion in is many times used in its contrapositive way. I.e., if we want tocheckthatasequenceisdivergent, wearesometimesabletospotanonconvergent subsequence, or, two subsequences that converge, but not to same the same limit. A prime example is the sequence (a n ) n N = (( 1) n ) n N. You have done question 3 where you have proved from first principles that (( 1) n ) n N is divergent. Using 1.2.7, we can re-confirm this: The subsequence (a 2n ) n N of (a n ) n N is the constant sequence 1, which converges to 1, whereas the subsequence (a 2n+1 ) n N of (a n ) n N is the constant sequence 1, which converges to -1. Hence by 1.2.7, the original sequence (a n ) n N must be divergent.
2 Scalar multiplication and addition of sequences Scalar multiplication and addition of sequences. It would be a tedious undertaking to prove convergence from first principles for all sequences. Instead we are looking for general principles which allow to deduce convergence (and to compute the limits) from known sequences after applying certain operations. In this section we introduce the simplest rules of this sort Scalar multiplication rule. If (a n ) n N is a convergent sequences and c R, then also (c a n ) n N is a convergent sequence and lim (c a n) = c lim a n. n n Proof. A special case here is c = 0. In this case the sequence (c a n ) n N is equal to the constant sequence of value 0, which converges to 0. We may therefore assume that c 0. We write r = lim n a n and we must show that (c a n ) n N converges to c r. According to the definition of convergence we have to pick some real number ε > 0 and we have to find some N N (depending on ε) such that for all n N, ( ) c a n c r < ε. How do we find such an N? Let us first write out what we know by our assumptions on the convergence of the given sequence: As (a n ) n N converges to r, there is some N 1 N (depending on ε) such that for all n N 1, It follows a n r < ε. c a n c r = c (a n r) = c a n r as c 0 < c ε. What now? Recall that we started with ε and we still have to find some N N such that for all n N, c a n c r < ε. On the other hand we were able to find the natural number N 1 (depending on ε) with the property that for all n N 1, c a n c r < c ε. The idea now is to apply the N 1 -argument for ε c instead of ε!! Then the argument above shows that we can find a natural number N 1 with the property that for all n N 1, ε c a n c r < c c = ε. Thus the natural number N, defined as N := N 1 (the symbol := stands for defined as ) that we found for ε c is suitable to solve our initial problem ( ) for ε Sum rule. If (a n ) n N, (b n ) n N are convergent sequences, then also their sum (a n +b n ) n N is a convergent sequence and lim n (a n +b n ) = lim n a n + lim n b n.
3 10 Sequences Proof. We write r = lim n a n, s = lim n b n and we must show that (a n + b n ) n N converges to r+s. According to the definition of convergence we have to pick some real number ε > 0 and we have to find some N N (depending on ε) such that for all n N, ( ) a n +b n (r +s) < ε. How do we find such an N? Let us first write out what we know by our assumptions on the convergence of the two given sequences: As (a n ) n N converges to r, there is some N 1 N (depending on ε) such that for all n N 1, a n r < ε. As (b n ) n N converges to s, there is some N 2 N (depending on ε) such that for all n N 2, b n s < ε. Hence if we take N 3 as the maximum of N 1 and N 2, then we know for all n N 3 : a n r < ε and b n s < ε. Now by the triangle inequality (cf (iii)), we get for all n N 3 : a n +b n (r +s) = (a n r)+(b n s) a n r + b n s ε+ε = 2ε. What now? Recall that we started with ε and we still have to find some N N such that for all n N, a n +b n (r +s) < ε. On the other hand we were able to find the natural number N 3 (depending on ε) with the property that for all n N 3, a n +b n (r +s) < 2ε. The idea now is to apply the N 3 -argument for ε 2 instead of ε!! Hence the argument above shows that we can find a natural number N 3 with the property that for all n N 3, a n +b n (r +s) < 2 ε 2 = ε. Thus the natural number N := N 3 that we found for ε 2 is suitable to solve our initial problem ( ) for ε.
4 Bounded and monotone sequences Bounded and monotone sequences Definition. For a subset S of R and an element r R we write S r if s r for every s S and S < r if s < r for every s S. Similarly the notation r S and r < S has to be understood. (i) Given a subset S of R we call every element r R with S r an upper bound for S. S is called bounded from above if there is an upper bound for S. Similarly every element r R with the property r S is called a lower bound for S. S is called bounded from below if there is a lower bound for S. S is called bounded if S is bounded from above and from below. For example the set of all rational numbers q with q 2 < 2 is bounded from above by 3 2 and bounded from below by 2. (ii) A sequence (a n ) n N is called bounded from above, bounded from below or bounded, if its value set has this property Proposition. Every convergent sequence is bounded. Proof. Let (a n ) n N be our convergent sequence with limit r. We choose ε = 1 and exploit the definition of convergence: Hence we know that there is some N N such that for all n N, a n r < 1; in other words r 1 < a n < r+1. Hence the set of all a n with n N is bounded above by r +1 and bounded below by r 1. However, the set {a 1,...,a N 1 } is finite and therefore bounded. Since the union of two bounded sets is bounded, the value set of (a n ) n N is bounded. The following cannot be proved and is an axiom for real numbers: Completeness axiom of R. Every nonempty subset S of R which has an upper bound, has a least upper bound Rules on least upper bounds and greatest lower bounds. IfS Risboundedfromabove,thenitsleastupperboundiscalledthesupremum of S and denoted by sup(s). For an element x R we have: x = sup(s) S x and for all r < x } there is some {{ s S with r < s }. S r A subset S R is bounded from below if and only if S := { s s S} is bounded from above, because S r r S for all r R. If this is the case, then S has a greatest lower bound, called the infimum of S, denoted by inf(s) and we have inf(s) = sup( S). Observe that the supremum of a bounded set S may be a member of S (e.g. if S = [0,1]) or may not be a member of S (e.g. if S = [0,1)). The existence of suprema of bounded sets is responsible for many constructions of real numbers. Most prominently, 2 is defined as sup{x Q x 2 < 2}.
5 12 Sequences Further, we pick up an example from section 1.1 now. Given a real number r > 0 and natural numbers n,m, we have a good intuition what r n m is: r n m is the m th root of r n. However, what shall we think of r 2? Or better: How is r 2 defined? We will answer this now with the aid of the axiom above. More generally, let r,p be real numbers. If r 1 we define r p := sup{r x x Q and x p}. We extend this definition for 0 < r < 1 by r p = ( 1 r ) p and show that We have r p = inf{r x x Q and x p} : r p = sup{( 1 r )x x Q and x p} = = sup{r x x Q and x p} = = sup{r x x Q and x p} = see below = inf{r x x Q and x p}. The last equality needs justification. Since r < 1 it is clear that for all x,y Q with x p y we have r x r y. Hence u := sup{r x x Q and x p} inf{r x x Q and x p} =: v. We now show that v u = 1 (and therefore v = u). We know already 1 v u and we show that for every real number δ > 0 we have v u < 1+δ. Choose n N with 1+nδ > 1 r and choose rational numbers x 1 p x 2 with x 2 x 1 < 1 n. By definition of u and v we have v rx1 and u r x2. Therefore Finally ( 1 r ) 1 n < 1+δ since 1 r v u rx1 r x2 = (1 r )x2 x1 by choice of n < 1+nδ as 1 r >1 and x2 x1< 1 n < ( 1 r ) 1 n. from the Binomial Theorem (1+δ) n Definition. Let (a n ) n N be a sequence. (a) (a n ) n N is called strictly increasing if a 1 < a 2 < a 3 <... and strictly decreasing if a 1 > a 2 > a 3 >... (a n ) n N is called strictly monotone if it is strictly increasing or strictly decreasing. (b) (a n ) n N is called increasing if a 1 a 2 a 3... and decreasing if a 1 a 2 a 3... (a n ) n N is called monotone if it is increasing or decreasing. The next theorem is a powerful tool to compute limits. We ll see examples shortly.
6 Bounded and monotone sequences Monotone convergence theorem. Every monotone and bounded sequence has a limit. Proof. Let (a n ) n N be our monotone sequence. We first assume that (a n ) n N is increasing, i.e. a 1 a 2 a 3... By assumption, (a n ) n N is bounded from above, which means that the value set A = {a n n N} is bounded from above. By the completeness axiom of R, A has a supremum s and we claim that s is the limit of (a n ) n N. To see this, take ε > 0. Since s is the least upper bound of A, s ε is not an upper bound of A, i.e. there is some N N with s ε < a N. Now if n N, then as (a n) n N is increasing as s=supa s ε < a N a n s. In particular a n s ε for all n N as required. It remains to do the case when (a n ) n N is decreasing. However, in this case the sequence ( a n ) n N is increasing (and again bounded). By what we have shown ( a n ) n N has a limit and by the scalar multiplication rule, also (a n ) n N has a limit.
Postulate 2 [Order Axioms] in WRW the usual rules for inequalities
Number Systems N 1,2,3,... the positive integers Z 3, 2, 1,0,1,2,3,... the integers Q p q : p,q Z with q 0 the rational numbers R {numbers expressible by finite or unending decimal expansions} makes sense
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 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 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 informationMATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:
MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is
More informationA lower bound for X is an element z F such that
Math 316, Intro to Analysis Completeness. Definition 1 (Upper bounds). Let F be an ordered field. For a subset X F an upper bound for X is an element y F such that A lower bound for X is an element z F
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 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 informationMAS221 Analysis Semester Chapter 2 problems
MAS221 Analysis Semester 1 2018-19 Chapter 2 problems 20. Consider the sequence (a n ), with general term a n = 1 + 3. Can you n guess the limit l of this sequence? (a) Verify that your guess is plausible
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 informationA lower bound for X is an element z F such that
Math 316, Intro to Analysis Completeness. Definition 1 (Upper bounds). Let F be an ordered field. For a subset X F an upper bound for X is an element y F such that A lower bound for X is an element z F
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 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 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 informationPrinciples of Real Analysis I Fall I. The Real Number System
21-355 Principles of Real Analysis I Fall 2004 I. The Real Number System The main goal of this course is to develop the theory of real-valued functions of one real variable in a systematic and rigorous
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 informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
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 informationMath 328 Course Notes
Math 328 Course Notes Ian Robertson March 3, 2006 3 Properties of C[0, 1]: Sup-norm and Completeness In this chapter we are going to examine the vector space of all continuous functions defined on the
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 informationa) Let x,y be Cauchy sequences in some metric space. Define d(x, y) = lim n d (x n, y n ). Show that this function is well-defined.
Problem 3) Remark: for this problem, if I write the notation lim x n, it should always be assumed that I mean lim n x n, and similarly if I write the notation lim x nk it should always be assumed 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 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 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 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 informationStructure of R. Chapter Algebraic and Order Properties of R
Chapter Structure of R We will re-assemble calculus by first making assumptions about the real numbers. All subsequent results will be rigorously derived from these assumptions. Most of the assumptions
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 informationExistence of a Limit on a Dense Set, and. Construction of Continuous Functions on Special Sets
Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets REU 2012 Recap: Definitions Definition Given a real-valued function f, the limit of f exists at a point c R
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 informationMATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7
MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7 Real Number Summary of terminology and theorems: Definition: (Supremum & infimum) A supremum (or least upper bound) of a non-empty
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 informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
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 informationEssential Background for Real Analysis I (MATH 5210)
Background Material 1 Essential Background for Real Analysis I (MATH 5210) Note. These notes contain several definitions, theorems, and examples from Analysis I (MATH 4217/5217) which you must know for
More informationLecture 2. Econ August 11
Lecture 2 Econ 2001 2015 August 11 Lecture 2 Outline 1 Fields 2 Vector Spaces 3 Real Numbers 4 Sup and Inf, Max and Min 5 Intermediate Value Theorem Announcements: - Friday s exam will be at 3pm, in WWPH
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 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 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 informationAxioms for the Real Number System
Axioms for the Real Number System Math 361 Fall 2003 Page 1 of 9 The Real Number System The real number system consists of four parts: 1. A set (R). We will call the elements of this set real numbers,
More informationRelations. Relations. Definition. Let A and B be sets.
Relations Relations. Definition. Let A and B be sets. A relation R from A to B is a subset R A B. If a A and b B, we write a R b if (a, b) R, and a /R b if (a, b) / R. A relation from A to A is called
More informationSolutions for Homework Assignment 2
Solutions for Homework Assignment 2 Problem 1. If a,b R, then a+b a + b. This fact is called the Triangle Inequality. By using the Triangle Inequality, prove that a b a b for all a,b R. Solution. To prove
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 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 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 informationImportant Properties of R
Chapter 2 Important Properties of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference
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 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 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 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 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 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 informationIntroduction to Mathematical Analysis I. Second Edition. Beatriz Lafferriere Gerardo Lafferriere Nguyen Mau Nam
Introduction to Mathematical Analysis I Second Edition Beatriz Lafferriere Gerardo Lafferriere Nguyen Mau Nam Introduction to Mathematical Analysis I Second Edition Beatriz Lafferriere Gerardo Lafferriere
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 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 informationLecture Notes in Real Analysis Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay
Lecture Notes in Real Analysis 2010 Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay August 6, 2010 Lectures 1-3 (I-week) Lecture 1 Why real numbers? Example 1 Gaps in the
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 informationSuppose R is an ordered ring with positive elements P.
1. The real numbers. 1.1. Ordered rings. Definition 1.1. By an ordered commutative ring with unity we mean an ordered sextuple (R, +, 0,, 1, P ) such that (R, +, 0,, 1) is a commutative ring with unity
More informationProblem 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 information2 Sequences, Continuity, and Limits
2 Sequences, Continuity, and Limits In this chapter, we introduce the fundamental notions of continuity and limit of a real-valued function of two variables. As in ACICARA, the definitions as well as proofs
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 informationGeneral Notation. Exercises and Problems
Exercises and Problems The text contains both Exercises and Problems. The exercises are incorporated into the development of the theory in each section. Additional Problems appear at the end of most sections.
More informationMIDTERM REVIEW FOR MATH The limit
MIDTERM REVIEW FOR MATH 500 SHUANGLIN SHAO. The limit Define lim n a n = A: For any ε > 0, there exists N N such that for any n N, a n A < ε. The key in this definition is to realize that the choice of
More information0.1 Pointwise Convergence
2 General Notation 0.1 Pointwise Convergence Let {f k } k N be a sequence of functions on a set X, either complex-valued or extended real-valued. We say that f k converges pointwise to a function f if
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 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 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 informationChapter 1. Sets and Numbers
Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More information5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing.
5 Measure theory II 1. Charges (signed measures). Let (Ω, A) be a σ -algebra. A map φ: A R is called a charge, (or signed measure or σ -additive set function) if φ = φ(a j ) (5.1) A j for any disjoint
More informationThis exam contains 5 pages (including this cover page) and 4 questions. The total number of points is 100. Grade Table
MAT25-2 Summer Session 2 207 Practice Final August 24th, 207 Time Limit: Hour 40 Minutes Name: Instructor: Nathaniel Gallup This exam contains 5 pages (including this cover page) and 4 questions. The total
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 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 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 informationConvex analysis and profit/cost/support functions
Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m Convex analysts may give one of two
More informationThat is, there is an element
Section 3.1: Mathematical Induction Let N denote the set of natural numbers (positive integers). N = {1, 2, 3, 4, } Axiom: If S is a nonempty subset of N, then S has a least element. That is, there is
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 informationDescribing the Real Numbers
Describing the Real Numbers Anthony Várilly Math 25a, Fall 2001 1 Introduction The goal of these notes is to uniquely describe the real numbers by taking certain statements as axioms. This exercise might
More informationProblem List MATH 5143 Fall, 2013
Problem List MATH 5143 Fall, 2013 On any problem you may use the result of any previous problem (even if you were not able to do it) and any information given in class up to the moment the problem was
More information2. The Concept of Convergence: Ultrafilters and Nets
2. The Concept of Convergence: Ultrafilters and Nets NOTE: AS OF 2008, SOME OF THIS STUFF IS A BIT OUT- DATED AND HAS A FEW TYPOS. I WILL REVISE THIS MATE- RIAL SOMETIME. In this lecture we discuss two
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 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 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 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 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 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 informationAdvanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017
Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with
More informationSection 2.5 : The Completeness Axiom in R
Section 2.5 : The Completeness Axiom in R The rational numbers and real numbers are closely related. The set Q of rational numbers is countable and the set R of real numbers is not, and in this sense there
More informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics and Engineering Lecture notes for PDEs Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 The integration theory
More information2 Lebesgue integration
2 Lebesgue integration 1. Let (, A, µ) be a measure space. We will always assume that µ is complete, otherwise we first take its completion. The example to have in mind is the Lebesgue measure on R n,
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 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 informationMATH5011 Real Analysis I. Exercise 1 Suggested Solution
MATH5011 Real Analysis I Exercise 1 Suggested Solution Notations in the notes are used. (1) Show that every open set in R can be written as a countable union of mutually disjoint open intervals. Hint:
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 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 informationEcon Lecture 2. Outline
Econ 204 2010 Lecture 2 Outline 1. Cardinality (cont.) 2. Algebraic Structures: Fields and Vector Spaces 3. Axioms for R 4. Sup, Inf, and the Supremum Property 5. Intermediate Value Theorem 1 Cardinality
More informationHOMEWORK ASSIGNMENT 6
HOMEWORK ASSIGNMENT 6 DUE 15 MARCH, 2016 1) Suppose f, g : A R are uniformly continuous on A. Show that f + g is uniformly continuous on A. Solution First we note: In order to show that f + g is uniformly
More informationStudying Rudin s Principles of Mathematical Analysis Through Questions. August 4, 2008
Studying Rudin s Principles of Mathematical Analysis Through Questions Mesut B. Çakır c August 4, 2008 ii Contents 1 The Real and Complex Number Systems 3 1.1 Introduction............................................
More informationSolution. 1 Solution of Homework 7. Sangchul Lee. March 22, Problem 1.1
Solution Sangchul Lee March, 018 1 Solution of Homework 7 Problem 1.1 For a given k N, Consider two sequences (a n ) and (b n,k ) in R. Suppose that a n b n,k for all n,k N Show that limsup a n B k :=
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 145 LECTURE NOTES. Zhongwei Zhao. My Lecture Notes for MATH Fall
MATH 145 LECTURE NOTES Zhongwei Zhao My Lecture Notes for MATH 145 2016 Fall December 2016 Lecture 1, Sept. 9 Course Orientation and Organization About the Professor Stephen New MC 5419 Ext 35554 Email:
More information02. Measure and integral. 1. Borel-measurable functions and pointwise limits
(October 3, 2017) 02. Measure and integral Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2017-18/02 measure and integral.pdf]
More information