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REALANALYSISNOTES
(2009)Prof. Sizwe Mabizela
Department of Mathematics (Pure & Applied)
Rhodes University
Contents1 Logic and Methods of Proof1
1.1 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 1
1.2 Tautologies, Contradictionsand Equivalences . . . . . . .. . . . . . . . . . . . . . . . . 7
1.3 Open Sentences and Quantifiers . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 9
1.4 Methods of Proof in Mathematics . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 13
1.4.1 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 13
1.4.2 ContrapositiveMethod . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 14
1.4.3 ContradictionMethod . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 14
2 Sets and Functions16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 16
2.2 Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 19
2.3 Indexed families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 22
2.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 24
2.5 Cardinality: the size of a set . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 27
2.5.1 The Cantor-Schr¨oder-BernsteinTheorem . . . . . . . . . .. . . . . . . . . . . . 33
3 Real Numbers and their Properties35
3.1 Real Numbers as a Complete Ordered Field . . . . . . . . . . . . . .. . . . . . . . . . . 35
3.1.1 The Archimedean Property of the Real Numbers . . . . . . . .. . . . . . . . . . 38
3.2 Topology of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 40
3.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 46
4 Sequences of Real Numbers50
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 50
4.2 Algebra of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 58
4.2.1 Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 60
4.2.2 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 63
4.2.3 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 64
4.2.4 Limit Superior and Limit Inferior . . . . . . . . . . . . . . . . .. . . . . . . . . 67
4.2.5 Sequential Characterization of Closed Sets . . . . . . . .. . . . . . . . . . . . . 71
4.2.6 Sequential Compactness . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 71
5 Limits and Continuity72
5.1 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 72
5.1.1 Algebra of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 76
5.2 ContinuousFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 79
5.2.1 Uniform Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 85
5.2.2 ContinuousFunctions and Compact Sets . . . . . . . . . . . . .. . . . . . . . . 88
i2009REALANALYSIS
6 Riemann Integration91
6.1 Basic Definitions and Theorems . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 91
6.1.1 Properties of the Riemann Integral . . . . . . . . . . . . . . . .. . . . . . . . . . 98
7 Introduction to Metric Spaces105
7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
7.2 Open Sets, Closed Sets, and Bounded Sets . . . . . . . . . . . . . .. . . . . . . . . . . . 109
7.3 Convergence of Sequences in Metric Spaces . . . . . . . . . . . .. . . . . . . . . . . . . 113
7.3.1 Sequential Characterization of closed sets . . . . . . . .. . . . . . . . . . . . . . 114
7.3.2 Completeness in Metric Spaces . . . . . . . . . . . . . . . . . . . .. . . . . . . 114
7.4 Compactness in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 116
7.4.1 Sequential Compactness . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 117
iiChapter 1Logic and Methods of Proof
1.1 Logic
In this course you will be expected to read, understand and construct proofs. The purpose of these notes is
to teach you the language of Mathematics. Once you have understood the language of Mathematics, you will be able to communicate your ideas in a clear, coherent and comprehensible manner.1.1.1 Definition
Aproposition(orstatement) is a sentence that is either true or false (not both).1.1.2 Examples
[1] South Africa was beaten by New Zealand in the 2003 cricketworld cup. [2] February 17, 2003 was on a Tuesday. [3]3C6D11. [4] p2is irrational.
1.1.3 Examples
(Examples of non-propositions). [1] Jonty is handsome. [2] What is the date? [3] This statement is true. There are two types of propositions:atomicandcompoundpropositions. ?An atomic propositionis a propositionthat cannot be divided into smaller propositions. ?A compoundpropositionis a propositionthat has parts that are propositions. Compoundpropositions are built by usingconnectives.1.1.4 Examples
(Examples of atomic propositions). [1] John"s leg is broken. 12009REALANALYSIS
[2] Our universe is infinite. [3] 2 is a prime number. [4] There are infinitely many primes.1.1.5 Examples
(Examples of compound propositions). [1] Jim and Anne went to the movies. [2]3?7. [3]n2is odd whenevernis an odd integer. [4] If a function is differentiable, then it is continuous. [5] Iff0>0, thenfis increasing. [6] Iffis increasing andf0exists, thenf0>0. Let us look at some of the most commonly used connectives:NameEnglish nameSymbol
Conjunctionand^
Disjunctionor_
ImplicationIf ...then)
Biconditionalif and only if,
Negationnot:
One has to be careful when using everyday English words in Mathematics as they may not carry the same meaning in Mathematics as they do in everyday non-mathematical usage. One such word isor. In everyday parlance, the wordormeans that you have a choice of one thing or the other butnot both-exclusivedisjunction. In Mathematics, on the other hand, the wordorstands for aninclusivedisjunction,
i.e., you have a choice of one thing or the other or both. We shall use the capital lettersP;Q;R; :::to denote atomic propositions.1.1.6 Examples
(Using symbols to represent compound statements). [1] If Lucille has credit for MAT 1E1 and MAT1E2, then she cannot get credit for MAT101. LetPstand for the statement Lucille has credit for MAT 1E1",Qstand for the statement Lucille has credit for MAT 1E2", andRstand for the statement Lucille can get credit for MAT 101." Then the above statement can represented symbolically as.P^Q/) :R. [2] If Lucille has credit for MAM100W or has credit for MAM105H and MAM106H, then she doMAM200W.
LetPstand for the statement Lucille has credit for MAM100W",Qstand for the state- ment Lucille has credit for MAM105H",Rstand for the statement Lucille has credit for MAM106H", andSstand for the statement Lucille can do MAM200W." Then the above statement can represented symbolically asOEP_.Q^R/?)S. [3] Either you pay your rent or I will kick you out of the apartment. 22009REALANALYSIS
LetPstand for the statement You pay your rent", andQstand for the statement I will kick you out of the apartment."Then the above statement can represented symbolically asP_Q. [4] Joe will leave home and not come back again. LetPstand for the statement Joe will leave home", andQstand for the statement Joe will come back again." Then the above statement can represented symbolically asP^ :Q. [5] The lights are on if and only if either John or Mary is at home. LetPstand for the statement The lights are on",Qstand for the statement John is at home", andSstand for the statement Mary is at home." Then the above statement can represented symbolically asP,.Q_S/. Atruth tableis a convenient device to specify all of the possible truth values of a given atomic orcompound proposition. We use truth tables to determine the truth or falsity of a compound proposition
based on the truth or falsityof its constituent atomic propositions.When we evaluate the truth or falsity of a statement, we assign to it one of the labelsTfor true" and
Ffor false". We also use1for true" and 0 for false". Let us construct truth tables for the above connectives. [1]Conjunction: LetPandQbe two propositions. The propositionP^Qis called the conjunctionof PandQ. The propositionP^Qis true if and only if both atomic propositionsPandQare true. In other words, if either or both atomic propositionsPandQare false, then the conjunctionP^Q is false.P Q P^Q 1 11 1 0 0 0 1 0 0 0 01.1.7 Examples
P: Cape Town is in the Western Cape and
p3is irrational.
Q:p5<3andf.x/D jxjis differentiable atxD0.
R: Harare is the capital of Botswana andf.x/Dcosxis continuous onR.S:?2 Only P is true; all the others are false.
[2]Disjunction: LetPandQbe two propositions. The propositionP_Qis called the disjunction of PandQ. The propositionP_Qis true if and only if at least one of the atomic propositionsPorQ is true. P Q P_Q 1 11 1 0 1 0 1 1 0 0 0 It is clear from this truth table that the propositionP_Qwill be false only whenbothPandQare false. 3 2009REALANALYSIS
1.1.8 Examples
(a)? >2or?is an irrational number. (b)? >2or?is a rational number. (c)? <2or?is an irrational number. (d)? <2or?is a rational number. All these propositions,except (d), are true.
[3]Implication: LetPandQbe two propositions. The propositionP)Qis referred to as acon- ditionalproposition. It simply means thatPimpliesQ. In the statementP)Q,Pis called the hypothesis(orantecedent or condition)andQis called theconclusion(orconsequent). There are various ways of stating thatPimpliesQ:
?IfP, thenQ. ?QifP. ?Pis sufficient forQ. ?Qis necessary forP. ?Ponly ifQ. ?QwheneverP. P Q P)Q 1 11 1 0 0 0 1 1 0 0 1 It is clear from this truth table that the propositionP)Qwill be false only whenPis true andQ is false. In order to have some appreciation of whythe above truthtable is reasonable, consider the following: If you pass MAM200W exam, I will buy you a cell-phone. LetP: You pass MAM200W exam.
LetQ: I will buy you a cell-phone.
At the end of MAM200W exam, there are various scenarios that may arise. (a) You have passed MAM200W exam and thenI buyyou a cell-phone. You will be happyand feel that I was tellingthe truth . ThereforeP)Qis true. (b) You have passed MAM200W exam but I refuse to buy you a cell-phone. You will feel cheated and lied to. ThereforeP)Qis false. (c) You have failed MAM200W, but I still buy you a cell-phone.You are unlikelyto question that, are you? We did not cover this contingency in my conditional statement. (d) You have failed MAM200W and, consequently I do not buy youa cell-phone. You will not feel that I have been unfair to you and that I have not kept my promise. 1.1.9 Examples
(a) If? >2, then?is an irrational number. 4 2009REALANALYSIS
(b) If? >2, then?is a rational number. (c) If? <2, then?is an irrational number. (d) If? <2, then?is a rational number. All these propositions, except (b), are true.
1.1.10 Definition
LetPandQbe propositions. Theconverseof the propositionP)Qis the propositionQ)P. 1.1.11 Examples
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