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1.1. Notation and Set TheoryIRA

Sets are the most basic building blocks in mathematics, and it is in fact not easy to give a

precise definition of the mathematical object set. Once sets are introduced, however, onecan compare them, define operations similar to addition and multiplication on them, and

use them to define new objects such as various kinds of number systems. In fact, most of

the topics in modern analysis are ultimately based on sets.

herefore, it is good to have a basic understanding of sets, and we will review a few

elementary facts in this section. !ost, if not all, of this section should be familiar and its

main purpose is to define the basic notation so that there will be no confusion in the

remainder of this te"t.

Definition 1.1.1: Sets and Operations on Sets

A set is a collection of objects chosen from some universe. he universe is

usually understood from the conte"t. Sets are denoted by capital, bold letters or

curly brackets.

A B# Ais a subsetof Bmeans that every element in Ais also contained

in B.

A B# A union Bis the set of all elements that are either in Aor in Bor

in both.

A B# A intersection Bis the set of all elements that are in both sets A

and B.

A \ B# A minus Bare all elements from Athat are not in B.

comp$A%# he complementof Aconsists of all elements that are not inA.

wo sets are disjointif A B& 0$the empty set%

wo sets Aand Bare equalif A Band B A

he most commonly used sets are the sets of natural numbers, integers, rational and

real numbers, and the empty set. hey are usually denoted by these symbols#

N& '(, ), *, +, ... & natural numbers $sometimes - is considered part of the

natural numbers as well%

Z& '... *, ), (, -, (, ), *, ... & integers

Q& 'p / 0 # p, 0 1 $read as 2all number p / 0, such that p and 0 are elements of

Z2% & rational numbers R& real numbers

0& empty set $the set that contains no elements%

All of the number systems $e"cept the natural numbers% will be defined in a

mathematically precise way in later sections. 3irst, some e"amples#

Examples 1.1.2:

4efine the following sets# E& 'x: x = 2nfor n N, O& 'x: x = 2n - 1

for n N, A& 'x R# + 5x5 *, B& 'x R # ( 5x5 6, and I& '

x R#x2= -2. hen#

(. 7hat, in words, are the sets E, O, and I8

). 3ind A B, A B, A9 B, comp$A%.

*. 3ind O E, O I, comp$I%.

Sets can be combined using the above operations much like adding and multiplying

numbers. 3amiliar laws such as associative, commutative, and distributive laws will be true

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for sets as well. As an e"ample, the ne"t result will illustrate the distributive law: other

laws are left as e"ercises.

Proposition 1.1.3: Distributive Law for Sets

A $B % & $A B% $A %

A (B C) = (A B) (A C)

;roof

!any results in set theory can be illustrated using onetheless,

before an actual proof is developed, it is first necessary to form a mental picture of the

assumptions, conclusions, and implications of a theorem. 3or this process a

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#roo$ by ontradiction# In this type of proof one assumes that the proposition $i.e.

what one actually would like to proof% is false. hen one derives a contradiction,

i.e. a logical impossibility. If that can be accomplished, then one has shown that the

negation of a statement will result in an illogical situation. =ence, the original

statement must be true.

Examples 1.1.$:

;rove that when two even integers are multiplied, the result is an even

integer, and when two odd integers are multiplied, the result is an odd

integer.

;rove that if the s0uare of a number is an even integer, then the

original number must also be an even integer. $ry a proof by

contradiction%

Euclid%s !"eoremstates that there is no largest prime. A proof bycontradiction would start out by assuming that the statement is false, i.e.

there is a largest prime. he advantage now is that if there was a largest

prime, there would be only finitely many primes. his seems easier to

handle than the original statement which implies the e"istence of

infinitely many primes. 3inish the proof.

1.2. Relations and FunctionsIRA

After introducing some of the basic elements of set theory $sets%, we will move on to thesecond most elementary concept, the concept of relations and functions.

Definition 1.2.1: %elation

et Aand Bbe two sets. A relationbetween Aand Bis a collection of ordered

pairs (a, b) such that a A and b B. Often we use the notation a ~ b to

indicated that aand bare related, rather then the order pair notation (a, b).

>ote that this does not mean that eachelement from Aneeds to be associated with one $or

more% elements from B. It is sufficient ifsomeassociations between elements of Aand B

are defined. In contrast, there is the definition of a function#

Definition 1.2.2: &un'tion( Domain( and %an#e

et Aand Bbe two sets. A $unctionffrom Ato Bis a relation between Aand B

such that for each a Athere is one and only one associated b B. he set Ais

called the domainof the function, Bis called its range.

Often a function is denoted asy = f(x)or simply f(x), indicatingthe relation { (x, f(x)) }.

Examples 1.2.3:

et A & '(, ), *, +, B & '(+, 6, )*+, & 'a, b, c, and R & real

numbers. 4efine the following relations#

(. ris the relation between Aand Bthat associates the pairs ( C )*+,) C 6, * C (+, + C )*+, ) C )*+

). f is the relation between A and that relates the pairs {(1,c),

(2,b), (3,a), (4,b)}

*. g is the relation between A and consisting of the associations

{(1,a), (2,a), (3,a)}

+. h is the relation between R and itself consisting of pairs

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{(x,sin(x))}

7hich of those relations are functions 8

he outcomes of a function $i.e. the elements from the range associated to elements in the

domain% do not only depend on the rule of the function $such as xis associated withsin(x)%

but also on the domain of the function. herefore, we need to specify those outcomes that

are possible for a given rule and a given domain#

Definition 1.2.!: )ma#e and Preima#e

et Aand Bbe two sets andfa function from A to B. hen the imageof f is

defined as

imag(f)& 'b B# there is an a Awithf(a) = b.

et A and Bbe two sets and fa function from A to B. If is a subset of the

range Bthen the preimage, or in&erse image, of under the functionfis the set

defined as f -1$% & 'x A#f(x)

As an e"ample, consider the following functions#

Example 1.2.$:

etf(x) = 0if xis rational andf(x) = 1if xis irrational. his function is

called 4irichletDs 3unction. he range forfis R.

o 3ind the image of the domain of the 4irichlet 3unction when#

(. the domain offis E

). the domain offis R

*. the domain offis F-, (G $the closed interval between - and

(%

o 7hat is the preimage of R8 7hat is the preimage of F(/),

(/)G 8

etf(x) = x2, with domain and range being R. hen use the graph of the

function to determine#

1. 7hat is the image of F-, )G and the preimage of F(, +G 8

2. 3ind the image and the preimage of F), )G.

3unctions can be classified into three groups# those for which every element in the image

has one preimage, those for which the range is the same as the image, and those whichhave both of these properties. Accordingly, we make the following definitions#

Definition 1.2.*: One+one( Onto( ,i-e'tion

A function ffrom A to Bis called one to one$or one one% if wheneverf(a) =

f(b)then a = b. Such functions are also called injections.

A function ffrom Ato Bis called ontoif for all b in Bthere is an ain Asuch

thatf(a) = b. Such functions are also called surjections.

A function f from A to Bis called a bijectionif it is one to one and onto, i.e.

bijections are functions that are injective and surjective.

Examples 1.2.:

If the graph of a function is known, how can you decide whether a

function is onetoone $injective% or onto $surjective% 8

7hich of the following functions are oneone, onto, or bijections 8

he domain for all functions is R.

(. f(x) = 2x +

). g(x) = arctan(x)

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*. g(x) = sin(x)

+. h(x) = 2x3+ x2- !x + "

1.3. Equivalence Relations and ClassesIRA

Hust as there were different classes of functions $bijections, injections, and surjections%,

there are also special classes of relations. One of the most useful kind of relation $besides

functions, which of course are also relations% are those called e0uivalence relations.

Definition 1.3.1: E/uivalen'e %elation

et 'be a set and r a relation between 'and itself. 7e call r an equi&alence

relationon 'if r has the following three properties#

(. Re$le(i&ity# very element of 'is related to itself). 'ymmetry# Ifsis related to tthen tis related tos

*. !ransiti&ity# Ifsis related to tand tis related to #, then s is related to #.

Examples 1.3.2:

et A & '(, ), *, + and B& 'a, b, c and define the following two

relations#

(. r : { (a,a), (b,b), (a,b), (b,a) }

). s :( C (, ) C ), * C *, + C +, ( C +, + C (, ) C +, + C )

7hich one is an e0uivalence relation, if any 8At first glance e0uivalence relations seem to be too abstract to be useful. =owever, just the

opposite is the case. Jecause they are defined in an abstract fashion, e0uivalent relations

can be utiliKed in many different situations. In fact, they can be used to define such basic

objects as the integers, the rational numbers, and the real numbers.

he main result about an e0uivalence relation on a set Ais that it induces a partition of A

into disjoint sets. his property is the one that will allow us to define new mathematical

objects based on old ones in the ne"t section.

Teorem 1.3.3: E/uivalen'e 0lasses

et rbe an e0uivalence relation on a set A. hen Acan be written as a union of

disjoint sets with the following properties#

(. If a, bare in Athen a ~ bif and only if aand bare in the same set

). he subsets are nonempty and pairwise disjoint.

he sets are called equi&alenceclasses.

;roof

Example 1.3.!:

Lonsider the set Zof all integers. 4efine a relation rby saying thatx

and$are related if their difference$ - xis divisible by ). hen

(. Lheck that this relation is an e0uivalence relation). 3ind the two e0uivalence classes, and name them appropriately.

*. =ow would you add these e0uivalence classes, if at all 8

7hat kind of e0uivalence classes do you get when xand$are defined

to be related if their difference is divisible by m8 =ow could you add

those 8

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=ere is another, more complicated e"ample#

Example 1.3.$:

Lonsider the set R" R9 '$-,-% of all points in the plane minus the

origin. 4efine a relation between two points (x,$)and (x%, $%)by saying

that they are related if they are lying on the same straight line passing

through the origin. hen#

o Lheck that this relation is an e0uivalence relation and find a

graphical representation of all e0uivalence classes by picking an

appropriate member for each class.

he space of all e0uivalence classes obtained under this e0uivalence

relation is called projecti&e space.

!ore e"amples for e0uivalence relations and their resulting classes are given in the ne"t

section.

1.4. Natural Nu!ers" #nte$ers" and RationalNu!ers

IRA

In this section we will define some number systems based on those numbers that anyone is

familiar with# the natural numbers. In a course on logic and foundation even the natural

numbers can be defined rigorously. In this course, however, we will take the numbers '(,

), *, +, ... and their basic operations of addition and multiplication for granted. Actually,

the most basic properties of the natural numbers are called the ;eano A"ioms, and aredefined $being a"ioms they need not be derived% as follows#

Definition 1.!.1: Peano xioms

( is a natural number

3or every natural numberxthere e"ists another natural numberx%called

the successor ofx.

1 & x%for every natural numberx$x%being the successor ofx%

Ifx% = $%thenx = $

If Qis a property such that#

(. ( has the property Q

). ifxhas property Qthenx%has property Q

then the property Qholds for all natural numbers.

he last property is called the ;rinciple of Induction and it will be treated in more detail in

the ne"t chapter. Right now we want to use the natural numbers to define a new number

system using e0uivalence classes, discussed in the previous section.

Teorem 1.!.2: Te )nte#ers

et Abe the set N" Nand define a relation r on N" Nby saying that (a,b)is

related to (a%,b%)if a + b% = a% + b. hen this relation is an e0uivalence relation.

If '(a,b)and '(a%, b%)denote the e0uivalence classes containing (a, b)and (a%,b%), respectively, and if we define addition and multiplication of those

e0uivalence classes as#

(. '(a,b) + '(a,b) = '(a + a, b + b)

). '(a,b) * '(a%, b%) = '(a * b% + b * a%, a * a% + b * b%)

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then these operations are welldefined and the resulting set of all e0uivalence

classes has all of the familiar properties of the integers $it therefore serves to

define the integers based only on the natural numbers%.

;roof

7hen defining operations on e0uivalent classes, one must prove that the operation is well

defined. hat means, because the operation is usually defined by picking particular

representatives of a class, one needs to show that the result is independent of these

particular representatives. Lheck the above proof for details.

he above proof is in fact rather abstract. he basic 0uestion, however, is# why would

anyone even get this idea of defining those classes and their operations, and what does this

really mean, if anything 8 In particular, why is the theorem entitled Mhe IntegersD 8 ry to

go through the few e"amples below#

Examples 1.!.3:

7hich elements are contained in the e0uivalence classes of, say, F$(,

)%G, F$-,-%G and of F$(, -%G 8 7hich of the pairs $(, N%, $N, (%, $(-, (+%, $6,*% are in the same e0uivalence classes 8

7hat do you get when adding, say, F$(,)%G F$+, P%G 8 =ow about

F$*,(%G F$(,*%G 8 7hat about multiplying F$N,+%G Q F$6, +%G and F$(,)%G Q

F$),(%G 8

Lan you think of a better notation for denoting the classes containing,

say, $N, 6% 8 =ow about the class containing $6, N% 8 4o you see why the

above theorem is called @he Integers@ 8

An obvious 0uestion about multiplication is# why did we not define

'(a, b) * '(a%, b%) = '(a * a%, b * b%)

It certainly looks a lot easier. Jut, keep in mind that all pair (a,b)are in the same class, if

their difference b - ais the same. =ence, we think of the class '(a,b)as represented by b -

aand of the class '(a%, b%)as the class represented by b% - a%. hen, in view of the fact that

(b - a) * (b% - a%) = b * b% + a * a% - (a * b% + a% * b)

our original definition of multiplication does make some sense.

Incidentally, now it is clear why multiplying two negative numbers together does give a

positive number# it is precisely the above operation on e0uivalent classes that induces this

interpretation. 3or e"ample#

$)% Q $+% &

his now has a precise mathematical interpretation.

$)% is a representative of the e0uivalence class of pair of natural numbers (a b)

$which we understand completely% whose difference b - a& ).

One such pair representing the whole class is, for e"ample, $+, )%.

o represent the class that might be denoted by +, we might choose the pair $, +%.

Jut then, according to our definition of multiplication, we have#

F$+, )%G Q F$, +%G & F $+ Q + ) Q % , $+ Q ) Q +%G & F$*), +-%G

and the appropriate notation for that class is +- *) & . =ence, we now understand

completely why $)% Q $+% & .

>e"t, we will define the rational numbers in much the same way, leaving the proof as an

e"ercise. o motivate the ne"t theorem, think about the following#

a b = a% b%if and only if a * b% = a% * b

a b + a% b% = (a b% + b * a%) (b * b%)

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a b * a% b% = (a * a%) (b * b%)

Teorem 1.!.!: Te %ationals

et Abe the set N" N ) *0+and define a relation ron N" N ) *0+by saying

that (a, b)is related to (a%, b%)if a * b% = a% * b. hen this relation is an

e0uivalence relation.

If '(a, b)and '(a%, b%)denotes the e0uivalence classes containing (a, b)and

(a%, b%), respectively, and if we define the operations

(. '(a, b) + '(a, b) = '(a * b + a% * b, b * b)

). '(a, b) * '(a%, b%) = '(a * a%, b * b%)

then these operations are welldefined and the resulting set of all e0uivalence

classes has all of the familiar properties of the rational numbers $it thereforeserves to define the rationals based only on the natural numbers%.

;roof

>ote that the second component of a pair of integers can not be Kero $otherwise the relation

would not be an e0uivalence relation%. As before, this will yield, in a mathematically

rigorous way, a new set of e0uivalence classes commonly called the @rational numbers@. he

individual e0uivalent classes '(a,b)are commonly denoted by the symbol a b, and are

often called a @fraction@. he re0uirement that the second component should not be Kero is

the familiar restriction on fractions that their denominator not be Kero.

As a matter of fact, the rational numbers are much nicer than the integers or the natural

numbers#

A natural number has no inverse with respect to addition or multiplication An integer has an inverse with respect to addition, but none with respect to

multiplication.

A rational number has an inverse with respect to both addition and multiplication.

Example 1.!.$:

So, why would anyone bother introducing more complicated

numbers, such as the real $or even comple"% numbers 8 3ind as many

reasons as you can.

2.1. Counta!le #n%nityIRA

One of the more obvious features of the three number systems N, Z, and Q that were

introduced in the previous chapter is that each contains infinitely many elements. Jefore

defining our ne"t $and last% number system, R, we want to take a closer look at how one

can handle @infinity@ in a mathematically precise way. 7e would like to be able to answer

0uestions like#

(. Are there more even than odd numbers 8

). Are there more even numbers than integers 8

*. Are there more rational numbers than negative integers 87hile most people would probably agree that there are just as many even than odd

numbers, it might be surprising that the answer to the last two 0uestions is no as well. All

of the sets mentioned have the same number albeit infinite of elements. he person who

first established a rigorous @theory of the infinite@ was . Lantor.

he basic idea when trying to count infinitely large $or otherwise difficult to count% sets

can roughly be described as follows#

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Suppose you are standing in an empty classroom, with a lot of students waiting to

get in. =ow could you know whether there are enough chairs for everyone8 ?ou

can not count the students, because they walk around too much. So, you simply let

in the students, one by one, and take a seat. If all the seats are taken, and no

students are left standing, then there was the same number of students as chairs.

his simple idea of matching two sets element by element is the basis for comparing two

sets of any siKe, finite or infinite. Since @matching elements from one set with those in

another set @ seems related to the concept of a function, we have arrived at the following

definition#

Definition 2.1.1: 0ardinalit

et A and B be two sets. 7e say that A and B have the same

cardinality if there is a bijection f from A to B. 7e write car() =

car(,).

If there e"ists a functionffrom Ato Bthat is injective $i.e. onetoone%we say that car() car(,).

If there e"ists a functionf from A to Bthat is surjective $i.e. onto% we

say that car() car(,).

;lease e"plain carefully what this definition has to do with the above idea of counting

students and chairs8

Examples 2.1.2:

7e can now answer 0uestions similar to the ones posed at the

beginning#

(. et Ebe the set of all even integers, Obe the set of odd integers.hen car(E) = car(O). 7hat is the bijection 8

). et Ebe the set of even integers, Zbe the set of all integers.

Again, car(E) = car(). Lan you find the bijection 8

*. et Nbe the set of natural numbers, Zbe the set of all integers.

7hich set, if any, has the bigger cardinality 8

Definition 2.1.3: 0ountable and 4n'ountable

If a set Ahas the same cardinality as N$the natural numbers%, then we say that

Ais countable. In other words, a set is countable if there is a bijection from

that set to N.

An alternate way to define countable is# if there is a way to enumeratethe

elements of a set, then the set has the same cardinality as Nand is called

countable.

A set that is infinite and not countable is called uncountable.

he second part of this definition is actually just rephrasing of what it means to have a

bijection from Nto a set A#

If a set Ais countable, there is a bijectionffrom Nto A. herefore, the elements

f(1), f(2), f(3), are all in A. Jut we can easily enumerate them, by putting them in

the following order# f(1)is the first element in A,f(2)is the second element in A,f(3)is the third one, and so on...

If a set Acan be enumerated, then there is a first element, a second element, a third

element, and so on. hen the function that assigns to each element of Aits position

in the enumeration process is a bijection between Aand Nand thus Ais countable

by definition.

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Jy the above e"amples, the set of even integers, odd integers, all positive and negative

integers are all countable.

>ote that there is a difference between finite and countable, but we will often use the word

countable to actually mean countable or finite $even though it is not proper%. =owever,

here is a nice result that distinguishes the finite from the infinite sets#Teorem 2.1.!: Dede5ind Teorem

A set 'is infinite if and only if there e"ists a proper subset Aof 'which has the

same cardinality as '.

;roof

Examples 2.1.$:

Tse 4edekind@s heorem to show that the set of integers Zand the

interval of real numbers between - and ), F-, )G, are both infinite $which

is of course not surprising%.

he surprising fact when dealing with countably infinite sets is that when combining twocountable sets one gets a new set that contains no more elements than each of the previous

sets. he ne"t result will illustrate that.

Proposition 2.1.*: 0ombinin# 0ountable Sets

very subset of a countable set is again countable $or finite%.

he set of all ordered pairs of positive integers is countable.

he countable union of countable sets is countable

he finite cross product of countable sets is countable.

;roof

hink about these propositions carefully. It seems to be contrary to ones beliefs. o seesome rather striking e"amples for the above propositions, consider the following#

Examples 2.1.:

he set of all rational numbers is countable.

he collection of all polynomials with integer coefficients is countable.

o prove this, follow these steps#

(. Show that all polynomials of a fi"ed degree n $with integer

coefficients% are countable by using the above result on finite

cross products.

). Show that all polynomials $with integer coefficients% arecountable by writing that set as a countable union of countable

sets.

2.2. &ncounta!le #n%nityIRA

he last section raises the 0uestion whether it is at all possible to have sets that contain

more than countably many elements. After all, the e"amples of infinite sets we encountered

so far were all countable. It was eorg Lantor who answered that 0uestion# not all infinite

sets are countable.Proposition 2.2.1: n 4n'ountable Set

he open interval $-, (% is uncountable.

;roof

>ote that this proposition assumes the e"istence of the real numbers. At this stage,

however, we have only defined the integers and rationals. 7e are not supposed to know

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anything about the real numbers. herefore, this proposition should from a strictly logical

point of view be rephrased#

if there are @real numbers@ in the interval $-, (% then there must be uncountably

many.

=owever, the real numbers, of course, do e"ist, and are thus uncountable. As for a moreelementary uncountable set, one could consider the following#

the set of all infinite se0uences of -@s and (@s is uncountable

he proof of this statement is similar to the above proposition, and is left as an e"ercise.

7hat about other familiar sets that are uncountable 8

Examples 2.2.2:

As for other candidates of uncountable sets, we might consider the

following#

(. 7hat is the cardinality of the open interval $(, (% 8

). 7hat is the cardinality of any open interval $a, b% 8

*. Is the set R of all real numbers countable or uncountable 8

>ow we know e"amples of countable sets $natural numbers, rational numbers% as well as

of uncountable sets $real numbers, any interval of real numbers%. Joth contain infinitely

many numbers, but, according to our counting convention via bijections, the reals actually

have a lot more numbers than the rationals.

Are there sets that contain even more elements than the real numbers 8 !ore generally,

given any set, is there a method for constructing another set from the given one that will

contain more elements than the original set 8 3or finite sets the answer is easy# just add one

more element that was not part of the original set. 3or countable sets, however, this does

not work, since the new set with one more element would again be countable.

Jut, there is indeed such a procedure leading to bigger and bigger sets#

Definition 2.2.3: Power Set

he power set of a given set 'is the set of all subsets of ', denoted by #-'..

Examples 2.2.!:

If '& '(,),*, then what is #-'.8 7hat is the power set of the set S

& '(, ), *, + 8 =ow many elements does the power set of S & '(, ), *,

+, N, P have 8

Show that car(P(S)) car (S) for any set ' $you donDt have to

proof strict ine0uality, only greater or e0ual%.Teorem 2.2.$: 0ardinalit of Power Sets

he cardinality of the power set #-'.is always bigger than the cardinality of '

for an set '.

;roof

Example 2.2.*: Lo#i'al )mpossibilities + Te Set of all Sets

7hen dealing with sets of sets, one has to be careful that one does not by

accident construct a logical impossibility. 3or e"ample, one might casually

define the following MsupersetD#

et 'be the set of all those sets which are not members of themselves.4o you see why this is an impossibility 8 Ask yourself whether the set 'is a

member of itself 8

Example 2.2.: 6ierar' of )nfinit + 0ardinal 7umbers

7e can now make a @hierarchy@ of infinity. he @smallest@ infinity is car(7).

he ne"t smallest infinity is car(%), then car(P(%)), then car(P(P(%))), and

so on. Jy the above theorem, we keep getting bigger and bigger @infinities@. he

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numbers car(S), where ' is any finite or infinite set, are called cardinal

numbers, and one can indeed establish rigorous rules for adding and

subtracting them.

ry to establish the following definitions for dealing with cardinalities#

(. 4efinition of a cardinal number

). Lomparing cardinal numbers

*. Addition of two cardinal numbers

Examples 2.2.8:

Tsing your above definitions, find the answers for the following

e"amples#

(. 7hat is car(7) + car(7)8

). 7hat is car(7) - car(7)8*. 7hat is car(%) + car(7)8

+. 7hat is car(%) + car(%)8

Example 2.2.9: Te 0ontinuum 6potesis

An important 0uestion when dealing with cardinal number is# is there a set '

whose cardinal number satisfies the strict ine0ualities

car(7) . car(S) . car(%)

7hat might be a possible candidate 8

In real analysis, we will $almost always% deal with finite sets, countable sets, or sets of the

same cardinality as card$R%. arger sets will almost never appear in this te"t, and the

e"istence of a set as described above will not be important to us.

In order to prove that two sets have the same cardinality one must find a bijection between

them. hat is often difficult, however. In particular, the difficulty in proving that a function

is a bijection is to show that it is surjective $i.e. onto%. On the other hand, it is usually easy

to find injective $i.e. onetoone% functions. herefore, thene"t theorem deals with e0ualityof cardinality if only onetoone functions can be found.

>ote that it should be the case that if you find a onetoone function from Ato B$i.e. each

element in A is matched with a different one from B, with possibly being unmatched

elements in Bleft over% and another onetoone function from Bto A$i.e. each element inBis matched with a different one from A, this time possibly leaving e"tra elements in A%,

then the two sets Aand Bshould contain the same number of elements.

his is indeed true, and is the content of the ne"t and last theorem#

Teorem 2.2.1: 0antor+,ernstein

et Aand Bbe two sets. If there e"ists a onetoone function ffrom Ato Band

another onetoone functiongfrom Bto A, then car() = car(,).

;roof

his theorem can be used to show, for e"ample, that R" Rhas the same cardinality as R

itself, which is one of the e"ercises. It can also be used to prove that Qis countable by

showing that Qand Z" Zhave the same cardinality $another e"ercise%.

2.3. The 'rinci(le o) #nductionIRA

In this section we will briefly review a common techni0ue for many mathematical proofs

called the ;rinciple of Induction. Jased on this principle there is a constructive method

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called Recursive 4efinition that is also used in several proofs. Joth principles, in fact, can

be applied to many wellordered set.

Definition 2.3.1: Ordered and ;ell+Ordered Set

A set 'is called partially orderedif there e"ists a relation r$usually denoted by

the symbol % between ' and itself such that the following conditions are

satisfied#

(. refle"ive# a afor any element ain '

). transitive# if a band b cthen a c

*. antisymmetric# if a band b athen a = b

A set 'is called orderedif it is partially ordered and every pair of elementsx

and$ from the set 'can be compared with each other via the partial ordering

relation.

A set 'is called /ell)orderedif it is an ordered set for which every nonempty

subset contains a smallest element.Examples 2.3.2:

4etermine which of the following sets and their ordering relations are

partially ordered, ordered, or wellordered#

(. 'is any set. 4efine a bif a = b

). 'is any set, and #$'% the power set of '. 4efine A Bif A B

*. 'is the set of real numbers between F-, (G. 4efine a bif ais less

than or e0ual to b$i.e. the @usual@ interpretation of the symbol %

+. 'is the set of real numbers between F-, (G. 4efine a b if a is

greater than or e0ual to b.

7hich of the following sets are wellordered 8(. he number systems N, Z,Q, or R8

). he set of all rational numbers in F-, (G 8

*. he set of positive rational numbers whose denominator e0uals

* 8

Teorem 2.3.3: )ndu'tion Prin'iple

et 'be a wellordered set with the additional property that every element e"cept

for the smallest one has an immediate predecessor. hen# if Qis a property such

that#

(. the smallest element of 'has the property Q). ifs 'has property Qthen the successor ofsalso has property Q

hen the property Qholds for every element in '

;roof

Recall the this is very similar to parts of the ;eano A"ioms, and it is easy to see that the

principle of induction applies to the wellordered set of natural numbers.

o use the principle of induction for the natural numbers one has to proceed in four steps#

(. 4efine a property that you believe to be true for some ordered set $such as N%

). Lheck if the property is true for the smallest number of your set $( for N%

*. Assume that property is true for an arbitrary element of your set $nfor N%

+. ;rove that the property is still true for the successor of that element $n+1for N%

Examples 2.3.!:

Tse induction to prove the following statements#

o he sum of the first npositive integers is n (n+1) 2.

o If a, b / 0, then (a + b) n an+ bnfor any positive integer n.

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Tse induction to prove Bernoulli%s inequality

o Ifx -1then (1 + x) n 1 + n xfor all n

Jefore stating a theorem whose proof is based on the induction principle, we should find

out why the additional property that every element e"cept the smallest one must have an

immediate predecessor is necessary for the induction principle#

Example 2.3.$:

(. he set of natural numbers, with the usual ordering, is wellordered,

and in addition every element e"cept of ( has an immediate predecessor.

>ow impose a different ordering labeled ..on the natural numbers#

o if nand mare both even, then define n .. mif n . m

o if nand mare both odd, then define n .. mif n . m

o if nis even and mis odd, we always define n .. m

Is the set of natural numbers, together with this new ordering ..well

ordered 8 4oes it have the property that every element has an immediate

predecessor 8

). Suppose the induction principle defined above does not contain the

assumption that every element e"cept for the smallest has an immediate

predecessor. hen show that it could be proved that every natural

number must be even $which is, of course, not true so the additional

assumption on the induction principle is necessary%.

A somewhat more complicated, but very useful theorem that can be proved by induction is

the binomial theorem#Teorem 2.3.*: ,inomial Teorem

et a and b be real numbers, and n a natural number. hen#

;roof

Jased on the Induction principle is the principle of Recursive 4efinition that is used

fre0uently in computer science.

Definition 2.3.: %e'ursive Definition

et 'be a set. If we define a function hfrom Nto 'as follows#

(. h(1)is a uni0uely defined element of '

). h(n)is defined via a formula that involves at most terms h()for 0 . .

n

hen this construction determines a uni0ue function hfrom Nto '.

Examples 2.3.8:

Jelow are two recursive definitions, only one of which is valid.

7hich one is the valid one 8

(. etx0= x1= 1and definexn= xn - 1+ xn - 2for all n / 1.). Select the following subset from the natural numbers#

x0& smallest element of N

xn& smallest element of '7- {x0, x1, x2, , xn + 1}

7hen first encountering proofs by induction, it seems that anything can be proved. It is

hard, in fact almost impossible, to find out why a particular property should be true when

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looking at an induction proof, and therefore, one might use induction to prove anything

$Incidentally, such proofs are often called nonconstructive proofs%.

Exer'ise 2.3.9:

ry to use induction to prove that the sum of the first ns0uare

numbers is e0ual to (n + 2)3. $his induction proof should fail, since

the statement is false or is it true 8%

=ere is a more elaborate e"ample of an invalid induction proof#

Example 2.3.1:

All birds are of the same color.

1#roo$1

he property is clearly true for n = 1, because Mone bird is of the same colorD.

Assume that nbirds are of the same color.

>ow take (n+1)birds. ;ut one aside. here are nbirds left, which, by assumption,

are of the same color. 3or simplicity, say they are all black. ;ut the one bird backinto the group, and take out another one. Again, there are n birds remaining, which

by assumption must have the same color. In particular, the bird that was taken out

in the first place must now have the same color as all the other birds, namely black.

he bird taken outside was also black. herefore, the n+1birds must all be black.

Jut that means that we have proved property Q for all natural numbers by

induction. ?et, the statement is obviously wrong. 7here does this proof actually

break down 8

A similar word of caution applies to Recursive 4efinitions. 7hile that principle can be

very useful, one has to be careful not to get into logical difficulties.

Example 2.3.11:

A classical e"ample for a recursive definition that does not work is the

parado" of the barber of Seville# he barber of Seville is that inhabitant

of Seville who shaves every man in Seville that does not shave himself.

he problem here is# who shaves the barber 8

o conclude, let@s prove two more @theorems@ via induction#

Examples 2.3.12: Sum of S/uares and 0ubes

;rove the following statements via induction#

(. he sum of the first nnumbers is e0ual to

). he sum of the first ns0uare numbers is e0ual to

*. he sum of the first ncubic numbers is e0ual to

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2.4. The Real Nu!er SysteIRA

In the previous chapter we have defined the integers and rational numbers based on thenatural numbers and e0uivalence relations. 7e have also used the real numbers as our

prime e"ample of an uncountable set. In this section we will actually define

mathematically correct the @real numbers@ and establish their most important properties.

here are actually several convenient ways to define R. wo possible methods of

construction are#

Lonstruction of Rvia 4edekindDs cuts

Lonstruction of Rclasses via e0uivalence of Lauchy se0uences .

Right now, however, it will be more important to describe those properties of Rthat we

will need for the remainder of this class.The rst question is: why do we need the real numbers !ren"t therationals good enough

Teorem 2.!.1: 7o S/uare %oots inote that we have not defined @ordered field@, but that is not so important for us right now.

he importance for us is that this property is one of the most basic properties of the real

numbers, and it distinguishes the real from the rational numbers $which do not have this

property%.

In order to prove this theorem we need to know what e"actly the real numbers are, and we

have indeed given two possible constructions at the beginning of this section.. =owever, it

is more important to #nerstanthese properties of R, and to know about the differences

between Rand the other number systems N, Z, and Q.7e can use this theorem to illustrate another property of the real numbers that makes them

more useful than the rational numbers#

Teorem 2.!.$: S/uare %oots in%

here is a positive real numberxsuch thatx2= 2

;roof

here are several other properties that will be of importance later on. wo of those are the

Archimedean and the 4ensity property. Again, as for the east Tpper Jound property, it is

more important to understand what these properties mean than to follow the proof e"actly.

Teorem 2.!.*: Properties of% and ow suppose that ewas rational, i.e. there are two positive integers aand bwith e = a b.

Lhoose n / b. hen, using the above representation, we have#

or multiplying both sides by nwe get#

Since n / b, the left side of this e0uation represents an integer, and hence n nis also an

integer. Jut we know that

so that if nis large enough the left side is less than (. Jut then n nmust be a positive

integer less than (, which is a contradiction.

>e"t we prove that9iis irrational, which is a lot harder to do. 7e need a preliminary

result#

Lemma 2.$.1:

4efine a functionfn(x) = . hen this function has the

following properties# 0 . fn(x) . 1 nfor 0 . x . 1

and are both integers

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#roo$Tsing the binomial theorem, we see that when the numerator of the function is

multiplied out, the lowest power ofxwill be n, and the highest power is 2n. herefore, the

function can be written as

fn(x) =

where all coefficients are integers. It is clear from this e"pression that = 0for ; .

nand for ; / 2n.!lso, loo$ing at the sum more carefully, we see that

...

%ut this implies that is an integer for any k.

&oreo'er, since

we also have

herefore is an integer for any ;.

F " G

Also note that < < . for nlarge enough $ n / 2a% and any a. >ow we can prove the

result of this chapter#

Teorem 2.$.2: Pi is irrational

9iis irrational

'roo),#e will pro'e that is not rational, which implies the assertion.uppose it was rational. Then there are two positi'e integers aand bwith

= a b

4efine the function

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ach of the factors

is an integer, by assumption on . 7e also know that and are integers as

well. =ence, (0)and (1)are both integers.

i*erentiating G(x)twice, we get:

he last term in this e0uation, the (2n+2)th derivative, is Kero, by the properties of the

functionf. Adding (x)and (x)we get#

>ow define another function

hen, using the above formula for (x) + (x), we have#

=

=

+ow we can use the second fundamental theorem of calculus toconclude:

= >(1) - >(0)

=

= ' (1) + (0)

hus, the integral

is an integer. Jut we also know that 0 . fn(x) . 1 nfor 0 . x . 1. herefore, estimating

the above integral, we get

0 . .

for 0 . x . 1. herefore, we can estimate our integral to get

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0 . . . 1

=ere we have used the fact that the last fraction approaches Kero if nis large enough. Jutknow we have a contradiction, because that integral was supposed to be an integer. Since

there is no positive integer less than 1, our assumption that was rational resulted in a

contradiction. =ence, must be irrational.

This pro'e is, admittedly, rather curious. !side from the assumption thatPiis rational (leading to the contradiction) the only other properties of Pithat are really in'ol'ed in this proof are that

sin( ) = 0, cos( ) = 1

and we need the defining properties of the trig. functions that

sin(x) = cos(x), cos(x) = - sin(x), sin(0) = 0, cos(0) = 1

hat does not make the proof much clearer, but it illustrates that some essential properties

of are indeed used along the way, and this proof will not work for any other number.aken from ?ac##s$)nd dition%, by@ichae A6ia;

;ublish or ;erish, Inc, (U-, pages *-6 *(-

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3.1. SequencesIRA

So far we have introduced sets as well as the number systems that we will use in this te"t.>e"t, we will study se0uences of numbers. Se0uences are, basically, countably many

numbers arranged in an order that may or may not e"hibit certain patterns. =ere is the

formal definition of a se0uence#

Definition 3.3.1: Se/uen'e

A sequenceof real numbers is a functionf:7 %. In other words, a

se0uence can be written asf(1), f(2), f(3), . Tsually, we will denote

such a se0uence by the symbol , where a= f().

3or e"ample, the se0uence 1, 12, 13, 14, 1, is written as . Beep in mind that

despite the strange notation, a se0uence can be thought of as an ordinary function. In many

cases that may not be the most e"pedient way to look at the situation. It is often easier to

simply look at a se0uence as a @list@ of numbers that may or may not e"hibit a certain

pattern.

#e now want to describe what the longterm beha'ior, or pattern, of asequence is, if any.

Definition 3.1.2: 0onver#en'e

A se0uence of real $or comple"% numbers is said to con&ergeto a real $or comple"% number cif for every / 0there is an integer

B / 0such that if / Bthen

< a- c < .

he number cis called the limit of the se0uence and we

sometimes write a c.

-f a sequence does not con'erge, then we saythat it diver$es.

Example 3.1.3:

Lonsider the se0uence . It converges to Kero.

;rove it.

he se0uence does not converge. ;rove it.

he se0uence converges to Kero ;rove it.

?onergentse0uences, in other words, e"hibit the behavior that they get closer and closer

to a particular number. >ote, however, that divergent se0uence can also have a regular

pattern, as in the second e"ample above. Jut it is convergent se0uences that will be

particularly useful to us right now.

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#e are going to establish se'eral properties of con'ergent sequences,most of which are probably familiar to you. &any proofs will use an argument as in the proof of the ne/t result. This type of argument is noteasy to get used to, but it will appear again and again, so that you

should try to get as familiar with it as you can.Proposition 3.1.!: 0onver#ent Se/uen'es are ,ounded

et be a convergent se0uence. hen the se0uence is bounded,

and the limit is uni0ue.

;roof

Example 3.1.$:

he 3ibonacci numbers are recursively defined asx1= 1,

x2= 1, and for all n / 2we setxn= xn - 1+ xn - 2. Show that the

se0uence of 3ibonacci numbers {1, 1, 2, 3, , }does notconverge.

Lonvergent se0uences can be manipulated on a term by term basis, just as one would

e"pect#

Proposition 3.1.*: l#ebra on 0onver#ent Se/uen'es

Suppose and are converging to aand b, respectively.

hen

(. heir sum is convergent to a + b, and the se0uences can be

added term by term.

). heir product is convergent to a * b, and the se0uences can bemultiplied term by term.

*. heir 0uotient is convergent to a b, provide that b & 0, and

the se0uences can be divided term by term $if the

denominators are not Kero%.

+. If an bnfor all n, then a b

;roof

This theorem states e/actly what you would e/pect to be true. The proofof it employs the standard tric$ of adding 0ero and using the triangle

inequality. Try to pro'e it on your own before loo$ing it up.+ote that the fourth statement is no longer true for strict inequalities. -nother words, there are con'ergent sequences with an< bnfor all n, butstrict inequality is no longer true for their limits. 1an you nd ane/ample #hile we now $now how to deal with con'ergent sequences, we stillneed an easy criteria that will tell us whether a sequence con'erges.

The ne/t proposition gi'es reasonable easy conditions, but will not tellus the actual limit of the con'ergent sequence.2irst, recall the following denitions:

Definition 3.1.: "onotoni'it

A se0uence is cae monotone in'reasin#if a + 1 afor a

A sequence is called monotone decreasingif

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aj aj + 1fo all j.

In other words, if every ne"t member of a se0uence is larger than the previous one, the

se0uence is growing, or monotone increasing. If the ne"t element is smaller than each

previous one, the se0uence is decreasing. 7hile this condition is easy to understand, thereare e0uivalent conditions that are often easier to check#

4onotone increasing

1 a + 1 a

2 a + 1- a 0

3 a + 1 a 1, if a/ 0

4onotone decreasing

1 a + 1 a

2 a + 1- a 0

3 a + 1 a 1, if a/ 0

Examples 3.1.8:

Is the se0uence monotone increasing or

decreasing 8

Is the se0uence monotone increasing or

decreasing 8

Is it true that a bounded se0uence converges 8 =ow aboutmonotone increasing se0uences 8

=ere is a very useful theorem to establish convergence of a given se0uence $without,

however, revealing the limit of the se0uence%# 3irst, we have to apply our concepts of

supremum and infimum to se0uences#

If a se0uence is bounded above, then c = s#6(x;)is finite. !oreover, given

any / 0, there e"ists at least one integer ;such thatx;/ c - , as illustrated in the

picture.

If a se0uence is bounded below, then c = inf(x;)is finite. !oreover, given

any / 0, there e"ists at least one integer ;such thatx;. c + , as illustrated in the

picture.

Proposition 3.1.9: "onotone Se/uen'es

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If is a monotone increasing se0uence that is bounded above,

then the se0uence must converge.

-f is a monotone decreasing sequence that isbounded below, then the sequence must con'erge.

;roof

3sing this result it is often easy to pro'e con'ergence of a sequence 4ustby showing that it is bounded and monotone. The downside is that thismethod will not re'eal the actual limit, 4ust pro'e that there isone.

Examples 3.1.1:

;rove that the se0uences and converge.

7hat is their limit8

4efinex1= band letxn= xn - 1 2for all n / 1. ;rove that

this se0uence converges for any number b. 7hat is the limit 8

et a / 0andx0/ 0and define the recursive se0uence

xn+1=12(xn+

a xn)

Show that this se0uence converges to the s0uare root of aregardless of the starting pointx0/ 0.

here is one more simple but useful theorem that can be used to find a limit if comparable

limits are known. he theorem states that if a se0uence is pinched in between two

convergent se0uences that converge to the same limit, then the se0uence in between must

also converge to the same limit.

Teorem 3.1.11: Te Pin'in# Teorem

Suppose {a}and {c}are two convergent se0uences such that im a=

im c= C. If a se0uence {b}has the property that

a b c

for all, then the se0uence {b}converges and im b= C.

;roof

Example 3.1.12:

Show that the se0uencesin(n) nand cos(n) nboth converge to

Kero.

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nteractie ea 5na$sis, ver. (.U.*$c% (UU+)---, Jert . 7achsmuth

3.2. Cauchy SequencesIRA

7hat is slightly annoying for the mathematician $in theory and in pra"is% is that we refer to

the limit of a se0uence in the definition of a convergent se0uence when that limit may not

be known at all. In fact, more often then not it is 0uite hard to determine the actual limit of

a se0uence.

#e would prefer to ha'e a denition which only includes the $nownelements of the particular sequence in question and does not rely on theun$nown limit. Therefore, we will introduce the following denition:

Definition 3.2.1: 0au' Se/uen'e

et be a se0uence of real $or comple"% numbers. 7e say that

the se0uence satisfies the auc"y criterion$or simply isauc"y% if

for each / 0there is an integerB / 0such that if, ; / Bthen

< a- a;< .

his definition states precisely what it means for the elements of a se0uence to get closer

together, and to stay close together. Of course, we want to know what the relation between

Lauchy se0uences and convergent se0uences is.

Teorem 3.2.2: 0ompleteness Teorem in %

et be a Lauchy se0uence of real numbers. hen the

se0uence is bounded.

5et be a sequence of real numbers. Thesequence is 1auchy if and only if it con'erges to somelimit a.

;roof

Thus, by considering 1auchy sequences instead of con'ergentsequences we do not need to refer to the un$nown limit of a sequence,and in e*ect both concepts are the same.+ote that the 1ompleteness Theorem not true if we consider onlyrational numbers. 2or e/ample, the sequence 6, 6.7, 6.76, 6.767, ...(con'ergent to the square root of 8) is 1auchy, but does not con'erge toa rational number. Therefore, the rational numbers are not complete, inthe sense that not e'ery 1auchy sequence of rational numberscon'erges to a rational number.9ence, the proof will ha'e to use that property which distinguishes the

reals from the rationals: the least upper bound property.

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3.3. Su!sequencesIRA

So far we have learned the basic definitions of a se0uence $a function from the naturalnumbers to the Reals%, the concept of convergence, and we have e"tended that concept to

one which does not presuppose the unknown limit of a se0uence $Lauchy se0uence%.

3nfortunately, howe'er, not all sequences con'erge. #e will nowintroduce some techniques for dealing with those sequences. The rst isto change the sequence into a con'ergent one (e/tract subsequences)and the second is to modify our concept of limit (li! su"and li! inf).

Definition 3.3.1: Subse/uen'e

et be a se0uence. 7hen we e"tract from this se0uence only

certain elements and drop the remaining ones we obtain a new

se0uences consisting of an infinite subset of the original se0uence.

hat se0uence is called a subsequenceand denoted by

One can e"tract infinitely many subse0uences from any given se0uence.

Examples 3.3.2:

ake the se0uence , which we have proved does

not converge. "tract every other member, starting with thefirst. 4oes this se0uence converge 8 7hat if we e"tract every

other member, starting with the second. 7hat do you get in

this case 8

ake the se0uence . "tract three different

subse0uences of your choice. 4o these subse0uences

converge 8 Is so, to what limit 8

he last e"ample is an indication of a general result#

Proposition 3.3.3: Subse/uen'es from 0onver#ent Se/uen'e

If is a convergent se0uence, then every subse0uence of that

se0uence converges to the same limit

-f is a sequence such that e'ery possiblesubsequence e/tracted from that sequences con'ergeto the same limit, then the original sequence alsocon'erges to that limit.

;roof

The ne/t statement is probably one on the most fundamental results ofbasic real analysis, and generali0es the abo'e proposition. -t alsoe/plains why subsequences can be useful, e'en if the original sequencedoes not con'erge.

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Teorem 3.3.!: ,ol=ano+;eierstrass

et be a se0uence of real numbers that is bounded. hen there

e"ists a subse0uence that converges.

;roof

Example 3.3.$:

4oes converge 8 4oes there e"ist a convergent

subse0uence 8 7hat is that subse0uence 8

In fact, the following is true# given any numberCbetween

( and (, it is possible to e"tract a subse0uence from the

se0uence that converges toC. his is difficult to

prove.

>e"t, we will broaden our concept of limits.

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3.4. -i Su( and -i #n)IRA

7hen dealing with se0uences there are two choices# the se0uence converges

the se0uence diverges

7hile we know how to deal with convergent se0uences, we don@t know much about

divergent se0uences. One possibility is to try and e"tract a convergent subse0uence, as

described in the last section. In particular, JolKano7eierstrass@ theorem can be useful in

case the original se0uence was bounded. =owever, we often would like to discuss the limit

of a se0uence without having to spend much time on investigating convergence, or

thinking about which subse0uence to e"tract. herefore, we need to broaden our concept of

limits to allow for the possibility of divergent se0uences.

Definition 3.!.1: Lim Sup and Lim )nf

et be a se0uence of real numbers. 4efine

5= inf{a, a + 1, a + 2, }

and let c = im (5). hen cis called the limit in$eriorof the se0uence

.

5et be a sequence of real numbers. ene

D= s#6{a, a + 1, a + 2, }

and let c = im (D). hen cis called the limit superiorof the

se0uence .

-n short, we ha'e:(. im inf(a) = im(5), where5= inf{a, a + 1, a + 2, }

). im s#6(a) = im(D), whereD= s#6{a, a + 1, a + 2, }

7hen trying to find lim sup and lim inf for a given se0uence, it is best to find the first few

5@s orD@s, respectively, and then to determine the limit of those. If you try to guess the

answer 0uickly, you might get confused between an ordinary supremum and the im s#6, or

the regular infimum and the im inf.

Examples 3.!.2:

7hat is inf,s#6, im infand im s#6for 8

7hat is inf,s#6, im infand im s#6for 8

7hat is inf,s#6, im infand im s#6for

7hile these limits are often somewhat counterintuitive, they have one very useful

property#

Proposition 3.!.3: Lim inf and Lim sup exist

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im s#6and im infalways e"ist $possibly infinite% for any se0uence

of real numbers.

;roof

-t is important to try to de'elop a more intuiti'e understanding aboutli! su"and li! inf. The ne/t results will attempt to ma$e these conceptssomewhat more clear.

Proposition 3.!.!: 0ara'teri=in# lim sup and lim inf

et be an arbitrary se0uence and let c = im s#6(a)and =

im inf(a). hen

(. there is a subse0uence converging to c

). there is a subse0uence converging to

*. im inf im s#6 cfor any subse0uence '

If cand are both finite, then# given any / 0there are arbitrary

largesuch that a/ c - and arbitrary large ;such that a;. +

;roof

! little bit more colloquial, we could say: 5picks out the greatest oEerbound for the truncated se0uences {a}. herefore5

tends to the smallest possible limit of any convergent subse0uence.

Similarly,Dpicks the smallest #66erbound of the truncated se0uences, and hence

tends to the greatest possible limit of any convergent subse0uence.

Lompare this with a similar statement about supremum and infimum.

Example 3.!.$

If is the se0uence of all rational numbers in the

interval '0, 1, enumerated in any way, find the im s#6and

im infof that se0uence.he final statement relates im s#6and im infwith our usual concept of limit.

Proposition 3.!.*: Lim sup( lim inf( and limit

If a se0uence {a}converges then

im s#6 a= im inf a= im a

Lonversely, if im s#6 a= im inf aare both finite then {a}

converges.

;roof

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3.+. S(ecial SequencesIRA

In this section will take a look at some se0uences that will appear again and again. ?oushould try to memoriKe all those se0uences and their convergence behavior.

#o/er 'equence

E(ponent 'equence

Root o$ n 'equence

n)t" Root 'equence

Binomial 'equence

Euler%s 'equence

E(ponential 'equence

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*e%nition 3.+.1, 'oer Sequence

#o/er 'equence# he convergence properties of the power se0uence

depends on the siKe of the base a#

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ase +1 A a A

Jy the above proof we know that < an

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*e%nition 3.+.2, E(onent Sequence

E(ponent 'equence# he convergence depends on the siKe of the

e"ponent a:

a / 0# the se0uence diverges to positive infinity

a = 0# the se0uence is constant

a . 0# the se0uence converges to -

back

/ou !roser is not 0avaena!led0 ...

E(ponent sequence /it" a > 2

E(onent sequence ith a = -2

'roo),

7rite n a= e a n(n). hen#

if a / 0then as napproaches infinity, the function ea n(n)approaches infinity as well

if a . 0then as napproaches infinity, the function ea n(n)approaches Kero

if a = 0then the se0uence is the constant se0uence, and hence convergent

Is this a good proof 8 Of course not, because at this stage we know nothing about the

e"ponential or logarithm function. So, we should come up with a better proof.

%ut actually we rst need to understand e/actly what nareally means: If ais an integer, then clearly nameans to multiply nwith itself atimes

If a = 68is a rational number, then n68means to multiply n6times, then take the

8root

It is unclear, however, what nameans if ais an irrational number. 3or e"ample,

what is n , or n 8

One way to define nafor all ais to resort to the e"ponential function#

na= ea n(n)

In that light, the original @proof@ was not bad after all, but of course we now need to know

how e"actly the e"ponential function is defined, and what its properties are before we can

continue. As it turns out, the e"ponential function is not easy to define properly. enerally

one can either define it as the inverse function of the nfunction, or via a power series.

!nother way to dene nafor abeing rational it to ta$e a sequence ofrational numbers ncon'erging to aand to dene naas the limit of the

sequence {nn}. There the problem is to show that this is welldened,i.e. if there are two sequences of rational numbers, the resulting limitwill be the same.-n either case, we will base our proof on the simple fact:

if6 / 0andx / $thenx6/ $6

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which seems clear enough and could be formally proved as soon as the e"ponential

function is introduced to properly definex6.+ow ta$e any positi'e number #and let nbe an integer bigger than #1$a.ince

n / F1a

we can raise both sides to the ath power to get

na/ F

which means sinceFwas arbitrary that the se0uence {na}is unbounded.The case a = %is clear (since n%= 1). The second case of a < %isrelated to the rst by ta$ing reciprocals (details are left as an e/ercise).ince we ha'e already pro'ed the rst case we are done.

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*e%nition 3.+.3, Root o) n Sequence

Root o$ n 'equence# his se0uence converges to (.

back

/ou !roser is not 0avaena!led0 ...

Root o$ n sequence

'roo),

If n / 1, then / 1. herefore, we can find numbers an/ 0such that

= 1 + anfor each n / 1

=ence, we can raise both sides to the nth power and use the Jinomial

theorem#

In particular, since all terms are positive, we obtain

Solving this for anwe obtain

0 an

Jut that implies that anconverges to Kero as napproaches to infinity, which

means, by the definition of anthat converges to ( as ngoes to infinity.

hat is what we wanted to prove.

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*e%nition 3.+.4, nth Root Sequence

n)t" Root 'equence# his se0uence converges to ( for any a / 0.

back

/ou !roser is not 0avaena!led0 ...

n)t" Root sequence /it" a > 3

'roo),

ase a @ 1

If a / 1, then for nlarge enough we have 1 . a . n. aking roots on both sides we

obtain

1 . .

Jut the righthand side approaches ( as ngoes to infinity by our statement of the

rootn se0uence. hen the se0uence { }must also approach (, being s0ueeKed

between ( on both sides $;inching theorem%.

ase A a A 1If 0 . a . 1, then (1a) / 1. Tsing the first part of this proof, the reciprocal of the

se0uence ' must converge to one, which implies the same for the original

se0uence.

Incidentally, if a = 0then we are dealing with the constant se0uence, and the limit is of

course e0ual to -.

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*e%nition 3.+.+, Binoial Sequence

Binomial 'equence# If b / 1then the se0uence converges to Kero

for any positive integer ;. In fact, this is still true if ;is replaced by any real

number.

back

/ou !roser is not 0avaena!led0 ...

Binomial sequence /it" 5 > 2and b > 1.3

'roo),

>ote that both numerator and denominator tend to infinity. Our goal will be to show that

the denominator grows faster than the ;th power of n, thereby @winning@ the race to infinity

and forcing the whole e"pression to tend to Kero.

The name of this sequence indicates that we might try to use thebinomial theorem for this. -ndeed, denexsuch that

b = 1 + x

Since b / 1we know thatx / 0. herefore, each term in the binomial theorem is positive,

and we can use the (;+1)st term of that theorem to estimate#

for any ;+1 n. et n = 2; + 1, or e0uivalently, ; = (n-1)2. hen n - ; = n - (n-1)2 =

(n+1)2 / n2, so that each of the e"pressions n, n-1, n-2, ..., n - ;is greater than n2.

=ence, we have that

Jut then, taking reciprocals, we have#

Jut this e"pression is true for all n / 2; + 1as well, so that, with ;fi"ed, we can take the

limit as napproaches infinity and the right hand side will approach Kero. Since the lefthand side is always greater than or e0ual to Kero, the limit of the binomial se0uence must

also be Kero.

-f kis replaced by any real number, see the e/ponential sequencetond out how nkcould be dened for rational and irrational 'alues of k.%ut perhaps some simple estimate will help if kis not an integer. etailsare left as an e/ercise.

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*e%nition 3.+., Euler Sequence

Euler%s 'equence# Lonverges to e ~

2!1G2G1G2G4H04233"02G!4!13 $uler@s number%. his se0uence

serves to define e.

back

/ou !roser is not 0avaena!led0 ...

Euler%s sequence

'roo),

7e will show that the se0uence is monotone increasing and bounded above. If that was

true, then it must converge. Its limit, by definition, will be called efor uler@s number.

ulers number eis irrational (in fact transcendental), and anappro/imation of eto ;< decimals is e &'.1'1'*%*'--%'*1-.2irst, we can use the binomial theorem to e/pand the e/pression

Similarly, we can replace nby n+1in this e"pression to obtain

he first e"pression has (n+1)terms, the second e"pression has (n+2)terms. ach of the

first (n+1)terms of the second e"pression is greater than or e0ual to each of the (n+1)

terms of the first e"pression, because

Jut then the se0uence is monotone increasing, because we have shown that

- 0

>e"t, we need to show that the se0uence is bounded. Again, consider the e"pansion

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1 +

>ow we need to estimate the e"pression to finish the proof.

-f we dene /n= , then

so that, finally,

for a n

Jut then, putting everything together, we have shown that

1 + 1 + An 3

for all n. =ence, uler@s se0uence is bounded by * for all n.

Therefore, since the sequence is monotone increasing and bounded, itmust con'erge. #e already $now that the limit is less than or equal to ;.-n fact, the limit is appro/imately equal to8.6>8>6>8>7?@

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*e%nition 3.+.5, E(onential Sequence

E(ponential 'equence# Lonverges to the e"ponential function

ex= ex6(x)for any real numberx.

back

'roo),

7e will use a simple substitution to prove this. et

xn = 1#, or e0uivalently, n = # x

hen we have

Jut the term inside the s0uare brackets is uler@s se0uence, which converges to uler@s

number e. =ence, the whole e"pression converges to ex, as re0uired.

-n fact, we ha'e used a property relating to functions to ma$e this proof

wor$ correctly. #hat is that property -f we did not want to use functions, we could rst pro'e the statementforxbeing an integer. Then we could e/pand it to rational numbers, andthen, appro/imatingxby rational number, we could pro'e the nalresult.

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4.1. Series and Conver$enceIRA

So far we have learned about se0uences of numbers. >ow we will investigate what mayhappen when we add all terms of a se0uence together to form what will be called an

infinite series.

The old Bree$s already wondered about this, and actually did not ha'ethe tools to quite understand it This is illustrated by the old tale of!chilles and the Tortoise.

Example !.1.1: enoBs Paradox C'illes and te Tortoise

Achilles, a fast runner, was asked to race against a tortoise. Achilles

can run (- meters per second, the tortoise only N meter per second.

he track is (-- meters long. Achilles, being a fair sportsman, gives

the tortoise (- meter advantage. 7ho will win 8

Joth start running, with the tortoise being (- meters ahead.

After one second, Achilles has reached the spot where the tortoise started. he

tortoise, in turn, has run N meters.

Achilles runs again and reaches the spot the tortoise has just been. he tortoise, in

turn, has run ).N meters.

Achilles runs again to the spot where the tortoise has just been. he tortoise, in

turn, has run another (.)N meters ahead.

his continuous for a while, but whenever Achilles manages to reach the spot where thetortoise has just been a splitsecond ago, the tortoise has again covered a little bit of

distance, and is still ahead of Achilles. =ence, as hard as he tries, Achilles only manages to

cut the remaining distance in half each time, implying, of course, that Achilles can actually

never reach the tortoise. So, the tortoise wins the race, which does not make Achilles very

happy at all.

Ob'iously, this is not true, but where is the mista$e +ow lets return to mathematics. %efore we can deal with any newob4ects, we need to dene them:

Definition !.1.2: Series( Partial Sums( and 0onver#en'e

et { a n}be an infinite se0uence.

(. he formal e"pression is called an $infinite% series.

). 3orB = 1, 2, 3, the e"pression im An= is called the

N)t" partial sumof the series.

*. If im Ane"ists and is finite, the series is said to con&erge.

+. If im Andoes not e"ist or is infinite, the series is said to

di&erge.>ote that while a series is the result of an infinite addition which we do not yet know how

to handle each partial sum is the sum of finitely many terms only. =ence, the partial sums

form a se0uence, and we already know how to deal with se0uences.

Examples !.1.3:

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= 12 + 14 + 1G + 11" + is an infinite series.

he *rd, +th, and Nth partial sums, for e"ample, are,

respectively# -.6N, -.U*6N, and -.UP6N.o 4oes this series converge or diverge 8

= 1 + 12 + 13 + 14 + 1 + is another infinite

series, called harmonic series. he *rd, +th, and Nth partial

sums are, respectively# (.**, ).-**, and ).)**.

o 4oes this series converge or diverge 8

Actually, if a series contains positive and negative terms, many of them may cancel out

when being added together. =ence, there are different modes of convergence# one mode

that applies to series with positive terms, and another mode that applies to series whose

terms may be negative and positive.

Definition !.1.!: bsolute and 0onditional 0onver#en'e

A series con&erges absolutelyif the sum of the absolute

values converges.

! series conver$es conditionally, if it con'erges, butnot absolutely.

Examples !.1.$:

4oes the series converge absolutely, conditionally,

or not at all 8

4oes the series converge absolutely,

conditionally, or not at all 8

4oes the series converge absolutely,

conditionally, or not at all $this series is called alternating

harmonic series% 8

Show that if a series converges absolutely, it

converges in the ordinary sense. he converse is not true.

Londitionally convergent se0uences are rather difficult to work with. Several operations

that one would e"pect to be true do not hold for such series. he perhaps most strikinge"ample is the associative law. Since a + b = b + afor any two real numbers aand b,

positive or negative, one would e"pect also that changing the order of summation in a

series should have little effect on the outcome. =owever#

Teorem !.1.*: bsolute 0onver#en'e and %earran#ement

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et be an absolutely convergent series. hen any

rearrangement of terms in that series results in a new series that is

also absolutely convergent to the same limit.

5et be a conditionally con'ergent series. Then, forany real number cthere is a rearrangement of theseries such that the new resulting series will con'ergeto c.

;roof

It seems that conditionally convergent series contain a few surprises. As a concrete

e"ample, we can rearrange the alternating harmonic series so that it converges to, say, ).

Examples !.1.: %earran#in# te lternatin# 6armoni' Series

3ind a rearrangement of the alternating harmonic series

that is within -.--( of ), i.e. show a concrete

rearrangement of that series that is about to converge to the

number ).

3ind a rearrangement of the alternating harmonic series

that diverges to positive infinity.

Absolutely convergent series, however, behave just as one would e"pect.

Teorem !.1.8: l#ebra on Series

et and be two absolutely convergent series. hen#

(. he sum of the two series is again absolutely convergent. Its

limit is the sum of the limit of the two series.

). he difference of the two series is again absolutelyconvergent. Its limit is the difference of the limit of the two

series.

*. he product of the two series is again absolutely convergent.

Its limit is the product of the limit of the two series $ Lauchy

;roduct%.

;roof

7e will give one more rather abstract result on series before stating and proving easy to

use convergence criteria. he one result that is of more theoretical importance is

Teorem !.1.9: 0au' 0riteria for Series

he series converges if and only if for every / 0there is a

positive integerBsuch that if m / n / Bthen

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< < .

;roof

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4.2. Conver$ence TestsIRA

In this section we will list many of the better known tests for convergence or divergence ofseries, complete with proofs and e"amples. ?ou should memoriKe each and every one of

those tests. he most useful tests are marked with a start $Q%. Llick on the 0uestion marks

below to learn more about that particular test.

4ivergence est $Q%

Lomparison est

imit Lomparison est $Q%

Lauchy Londensation est

eometric Series est $Q%

p Series est $Q%

Root est

Ratio est $Q%

Abel@s Lonvergence est

Alternating Series est $Q%

Integral est

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4.3. S(ecial SeriesIRA

his is a brief listing of some of the most important series. ach of the convergence testsgiven in the previous section contains even more e"amples.

5eometric 'eries

converges absolutely for

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*iver$ence Test

If the series converges, then the se0uence converges to Kero.

0uivalently#

-f the sequence does no0con'erge to 0ero, the