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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

i

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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

ii

Chapter 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] p

2is 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. 1

2009REALANALYSIS

[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 do

MAM200W.

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. 2

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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 or

compound 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 0

1.1.7 Examples

P: Cape Town is in the Western Cape and

p

3is irrational.

Q:p

5<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

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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

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(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

(Examples of converse statements). (a) If it is cold, then the lake is frozen.

Converse: If the lake is frozen, then it is cold.

(b) Johny is happy if he is healthy.

Converse: If Johny happy, then he is healthy.

(c) If it rains, Zinzi does not take a walk. Converse: If Zinzi does not take a walk, then it rains.

The truth table of a propositionand its converse:

P Q

P)QQ)P

1 111 1 0 01 0 1 10 0 0 11 Note that the truthtables ofP)QandQ)Pare not the same. Consider the followingconditional propositionand its converse:

Proposition: If? >2, thenp

3is rational.

Converse: Ifp

3is rational, then? >2.

In this example the conditional statement is false whereas its converse is true. Hence this conditional

propositionand its converse are not equivalent. Consider the followingconditional propositionand its converse:

Proposition: If? >2, thenp

3is irrational.

Converse: Ifp

3is irrational, then? >2.

Here boththat conditionalpropositionand itsconverse aretrue. If, inthisexample, we letPstandfor the proposition“? >2" andQfor “p

3is irrational", then we have that bothP)QandQ)P

are true. [4]Biconditional Proposition: LetPandQbe propositions. The propositionP,Qis referred to as abiconditionalproposition. It simply means thatP)QandQ)P. It is called a “biconditional proposition"because it represents two conditional propositions. There are various ways of stating the propositionP,Q: ?Pif and only ifQ(also written asPiffQ). 5

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?PimpliesQandQimpliesP. ?Pis necessary and sufficient forQ. ?Qis necessary and sufficient forP. ?Pis equivalent toQ. P Q P,Q 1 11 1 0 0 0 1 0 0 0 1 Note that the statementP,Qis true precisely in the cases wherePandQare both true orPand

Qare both false.

[5]Negation: LetPbe a proposition. The proposition:P, meaning “notP", is used to denote the negation ofP. IfPis true, then:Pis false and vice versa. P :P 10 0 1

Let us construct a few more truth tables.

1.1.12 Examples

[1] LetPandQbe propositions. Construct a truth table for the proposition.P^Q/).P_Q/.

Solution:

P

QP^QP_QP^Q)P_Q

11111
1 0011 0 1011
0 0001 [2] LetP;QandRbe propositions. Construct the truth table for the proposition:.P^Q/_R.

Solution:

P

QRP^Q:.P^Q/:.P^Q/_R

111101

1 10100
1 01011
0 11011
1 00011 0 10011
0 01011
0 00011 6

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1.2 Tautologies, Contradictions and Equivalences

Some compound propositionsare always true while others arealways false.

1.2.1 Definition

A compound propositionis atautologyif it is always true regardless ofthe truthvalues of its atomic propo-

sitions. If, on the other hand, a compound propositionis always false regardless of its atomic propositions,

we say that such a propositionis acontradiction.

1.2.2 Example

The statementP_ :Pis always true while the statementP^ :Pis always false. P :PP_ :PP^ :P 1010
0 110

1.2.3 Remark

In a truth table, if a proposition is a tautology, then every line in its column will have 1 as its entry;

if a proposition is a contradiction, every line in its columnwill have 0 as its entry.

1.2.4 Definition

LetPandQbepropositions. ThecontrapositiveofthepropositionP)Qistheproposition:Q) :P.

1.2.5 Examples

(Examples of contrapositive statements). [1] If it is cold, then the lake is frozen. Contrapositive: If the lake is not frozen, then it is not cold. [2] If Johny is healthy, then he is happy. Contrapositive: If Johny not happy, then he is not healthy. [3] If it rains, Zinzi does not take a walk. Contrapositive: If Zinzi takes a walk, then it does not rain. DO NOT CONFUSE THE CONTRAPOSITIVE AND THE CONVERSE. Here is the difference: Converse: The hypothesisof a converse statement is the conclusion ofthe conditional statement and the conclusion of the converse statement is the hypothesisof the conditional statement. Contrapositive: The hypothesis of a contrapositive statement is thenegationof conclusion of the conditionalstatement and the conclusionof the contrapositivestatement is thenegationofhypothesis of the conditional statement.

1.2.6 Examples

[1] If Bronwyne lives in Cape Town, then she lives of South Africa. Converse: If Bronwyne lives in South Africa, then she lives in Cape Town. 7

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Contrapositive: If Bronwyne does not live in South Africa, then she does not live Cape Town. [2] If it is morning, then the sun is in the east. Converse: If the sun is in the east, then it is morning. Contrapositive: If the sun is not in the east, then it is not morning.

1.2.7 Definition

TwopropositionsPandQare saidtobelogicallyequivalent,writtenasP?Q, ifP,Qisa tautology. Logically equivalent statements have the same truth values.

1.2.8 Remark

When we write “P?Q", we basically say that propositionP means the sameas propositionQ. Here is an important example:P)Q? :Q) :P. That is, the conditionaland its contrapositivesay the same thing. P

QP)Q:P:Q:Q) :P.P)Q/,.:Q) :P/

1110011

1

000101

0

111011

0

011111

1.2.9 Theorem

LetP;QandRbe propositions. Then

(a):.P^Q/? :P_ :Q (b):.P_Q/? :P^ :Q (c):.P)Q/?P^ :Q (d)P)Q? :P_Q (e):.:P/?P (f)P_.Q^R/?.P_Q/^.P_R/ (g)P^.Q_R/?.P^Q/_.P^R/ (h).P_Q/_R?P_.Q_R/ (i).P^Q/^R?P^.Q^R/

Proof. (a):.P^Q/? :P_ :Q:

P

QP^Q:.P^Q/:P:Q:P_ :Q:.P^Q/,.:P_ :Q/

11100001

1

0010111

0

1011011

0

0011111

(c):.P)Q/?P^ :Q 8

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PQP)Q:.P)Q/:QP^ :Q:.P)Q/,.P^ :Q/

1110001

1

001111

0

110001

0

010101

Try to convince yourself that all the other statements are valid.

Let us analyze the following argument: If girls are blonde, they are popular with boys. Ugly girls are

unpopular with boys. Intellectual girls are ugly. Therefore blonde girls are not intellectual.

Is this argument valid?

Solution: Let us use letters and connectives to represent the above statement.

P: Girls are blonde.

Q: Girls are popular with boys.

R: Girls are ugly.

S: Girls are intellectual.

We can represent the above argument as follows:

P)Q;R) :Q;S)R:

SinceS)RandR) :Q, we can conclude thatS) :Q.

SinceP)Q, we have, by contrapositive, that:Q) :P. Hence,S) :P. Again, by contrapositive,P) :S, which says that “Blonde girlsare not intellectual." Therefore the argument is valid.

1.3 Open Sentences and Quantifiers

Inmathematics, onefrequentlycomes acrosssentences thatinvolveavariable. Forexample,x2C2x?3D0

is one such. The truth or falsity of this statement depends onthe value you assign for the variablex. For

quotesdbs_dbs10.pdfusesText_16

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