EQUIVALENCE CLASSES OF CAUCHY SEQUENCES

OF RATIONAL NUMBERS

APPROVEDi

L£.., Ma4or Professor

Minor Professor

\nl/i^u.. Director of the Department of Mathematics

Dean of" the' Graduate School

EQUIVALENCES CLASSES OF CAUCHX SEQUENCES

OF RATIONAL NUMBERS

THESIS

Presented to the Graduate council of the

North Texae State University in Partial

Fulfillment of the Requirements

For the Degrees of

MASfER OF ARTS

By

Linda Jane Darnell, B. A.

Denton, Texas

January, 1965

TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION 1

II. FIELD OP EQUIVALENCE CLASSES 2

III. ORDER AID COMPLETENESS OF THE CLASSES. . . .IS

I?. UN COUNT ABILITY OF THE CLASSES. 26

BIBLIOGRAPHY. 32

CHAPTER I

INTRODUCTION

The ©volution of the real number system from the

system of rational numbers m&j be obtained in a variety

of ways,. The Dedekind out method is due to the German

mathematician R, Dedekind, However, Georg Cantor's

approach, which is more analytic and 1®a® algebraic

than that of Dedekind, will be discussed in this thesis.

Cantor used the basic notion of Cauchy sequences of

rational numbers.

The purpose of this thesis is to define ©quivalence

classes of Cauchy sequences of rational numbers and the

operations of taking a sua and a product and then to

show that this systea is an uncountable, ordered,

complete field. In so doing, a mathematical system is

obtained which is isomorphic to the real number system.

All properties of the rational numbers and integer®

will be assumed throughout the thesis.

CHAPTER II

FIELD OF EQUIVALENCE CLASSES

Definition 2,1

x li a aequeno® if and only if x is & function on

the set of natural nustbers.

Definition 2,2

If x is a sequence, then x(n)»xat that 1ft, x n 1®

the n-th »#mber or term of the sequence x.

Definition 8.3

If i is i sequence, then xajx^, x2» x^, , • . f xn> • * • •

Definition 2.4

x 18 a sequence of rational numbers if and only

if x le a sequence and the range of x is a subset of

the set of rational numbers.

Definition 2,5

z is a Gauehy sequence of rational numbers if and

only if x Is a sequence of rational numbers, and for

everj positive rational number e there is a positive

integer I such that for every positive integer a,n >N,

I Im"xnl<te*

Definition 2,6

If each of x and y i® a Cauchy 8®qu®aee of rational

numbers, then x^y if and only if for every positive

rational number e there is a positive integer I such that

for every integer n>K» Un-Jnl^ e»

Theorem 2,1

If x is a Cauchy sequence of rational number®, then

Proof; Suppose e la s positive rational number.

For every positive integer n, l*n-xnl«0

<#* Hence x^x.

Theorem 2.2

If eaoh of x and y is a Cauchy sequence of rational

numbers and x-CVy, then y -^x.

Proof: Let e be a positive rational number. There

is a positive integer I suoh that for integers ti>M»

\ %-yBl •. But for integers n >1» lyn-*J« l*n*-ynl • •

Henee y^x.

Theorem 2,3

If each of x, y» and 2 is & Cauchy sequence of rational

numbers and x —y and y~u» then x^z.

Proof: Let e be a positive rational number, There

Is a positive integer such that for Integers n > % ,

Ix^-yJ g. There is a positive integer lg such that

for integers n>» 2, Let Haaaximua »2^,

then for integers n>N, [x^-y^l^l and I ^ |* Thu*»

for integers n>H,

^ I v y J + l v n

* • *

Hence x^«.

Definition, 2.7

If x is a Cauchy sequence of rational, nuabera, then

X*£jr|y is a Cauchy sequence of rational numbers and

y — x will to# called an equivalence class#

theorem 2.4

If A is an equivalence class and each of x,y Is in A,

then x~y.

Prooft Suppose a is the Cauchy sequence of rational

numbers such that A%fb|b is a Cauchy sequence of rational

numbers and b~a?. Since x,y£A, then x^e and y — a.

Since y-a» then a^y» Since x^a and a^y, then x—y #

Theorem 2.5

If each of 1 and B is an equivalence class and a it

in A and b is in B and a-^b, then AssB.

Proofs There is a sequence g in A such that

A»^x|x la a Cauchy sequence of rational numbers and x —gj.

There is a sequence h in B such that B*fy|y ia a Cauchy

sequence of rational numbers and y—hj. Slr.ce a^b and a

is in A, then b—a and a —g. Hence b^g and therefore

b in in A. Since a^b and b is in 1, then b—h. Hence

a—h # or a i® in B. Therefore AssB.

Rieorem 2,6

It x 1® a Cauchy sequence of rational numbers» then

ther® 1b a positive rational number d sueh that for every

positive integer n, |xn|^d.

froott Ihere ia a positive Integer 1 auoh that for

Integers u,»>If» j^-x^Ul, Let dsaaaxlmum f\xx\ , |x2|,

|^|# . • . * |xjf|» l*n+i 11 • & is a rational number.

Sow for positive integers n^N+1, )xn|«£d. For positive

integers n>»+l, then JxJ- 1*K+1I < 1*

Hencet jx J*: J d» Hence, for every positive

integer n, d.

Definition 2.S

If each of x and y is a sequence of rational numbers,

then *•; is a sequence z so that for every positive

Integer a,

Definition 2.9

If eath of x and y ia a sequence of rational numbers,

then icy i® a sequence t so that for every positive

integer a, anJ»xn»yn.

theorem 2.7

If each of x and y is a cauchy sequence of rational

nuabere» then so are x+y and x»y.

Proofi For every positive Integer n» Xn+yn and xn*¥n are rational numbers. Hence x+y and x*y are sequences

of rational numbers.

Let « b® a posi t ive ra t iona l number * There are

posi t ive integers and Sg *uch tha t fo r integer® a , n > N i ,

| x t t~xn | c | and for integers nt,n>N2, < \* Let

Hamaximun lg^ | then for integers m,n>N,

l< V7«M*ii*XB>I= I

as® •

Hence x+y i s Cauohy.

Let e be a posi t ive ra t iona l number. There are

posi t ive ra t iona l numbers Mid d2 such that fo r every

pos i t ive integer n , Jx^ < d^ and |yft | <id2. Let

dsaaxiaum fd1 # dg j . low ^ i s a posi t ive ra t iona l number.

There are posi t ive integers and 8g such that fo r

integers j x ^ x j 81X1(1 t o r Integer® m»n>lg >

I V y n I * jjy • Let Kamaximua £m1# Kg|. Now fo r integers

* 4 ' z r * a'h

" 5 * 5 s# *

Hencet x*y Is Cauchy.

Theorem 2.8

If eaeh of A and B Is an equivalence elass and eaeh

of v#xeA and each of y ,z£B, then v+y<£!x+z.

Proof: Since w,x£A and y,z€B, then w-^x and y-&%,

Let s be a positive rational number, There are positive

integers and N2 such that for every Integer n >%»

l*tt-*nl*§ and for every integer n >S2, |yn-zB|^|.

Let Haaaximum %!# then for every integer n> N»

I *„"% U | <»« l/n-^l | • » o w

I ("n+ynMvan)! - l< I

£ l*a"xnl*!ja""!1til

< 1 * I

I® 0 *

Hence, w+y^x+z.

Theorem 2*9

If each of A and B Is an equivalence ©lass and eaelt

of v,x6A and each of y»a6Bf then wy^x»i,

Proof j Since w,xeA and y,z€B, then w^x and ya? *.

Let e be a positive rational number» There are positive

rational numbers d^ and dg such that for every positive

integer n, & n d l*nl * d2* *** dcaaximum £%* dg^.

There are positive Integers and N2 such that for integer®

n>Ml* k n - x n | a n d for Integers n>Mg» |yn-zn|

Let K*anaxlmum fylt fgj. Then for integers n>N,

|wn-yn-xn»2nl»)wn^n-xn-yn+xn-yn-*n'anl

* |wn,yn-xn*ynU|%'yn-xn#anl

«|y n | ' l w n - * n | • U n l ' l Jn -%I

<3* z'*' 4* d * <yii 24 2d ® #

» * + *

s ®,

Hence, w*y—x»z«

Definition 2,10

If each of 1 and B is an equivalence ©lass and 0 is

an ©quivalence class so that there la a 8«q«inc@ a In A

and a sequence b in B so that a*b la in C, then 0 1® a

sum of A and B and will be denoted by A+BaC*

Theorem 2.10

If each of A and B is an equivalence class, then

A+B exists and is unique.

Proof: Since each of A and B is an equivalence

class, there is a sequence a in A and a sequence b in Bt

and a*b Is a Cauchy sequence of rational numbers} call

it c. Let Ca icjx 1® a Sauchy sequence of rational nuabers

and x—cjf, Hence A*B exists.

Suppose there are two equivalence classes, G and C*,

such that A*BeC and A+B=rC'; that is, there it a sequence

a in A and a sequence b in B such that a+t is in C; and

there is a sequence a* in A and a sequence b* in B such

that a'+b* is in C1 • But a+fc-a'+b', Since a*b is in C

and a'+b* is in C' and a+b^a'+b*» then CaC*.

Definition 2.11

If each of A and B Is an equivalence class and C is

an equivalence Glass ao that there is a sequence a in A

ana a sequence b in B so that a*b is in C, then 0 is a

product of A and 5 and will be denoted by A*B*C,

Theorem 2,11

If each of A &n«l B is an equivalence class, then

A*S exists and is unique*

Frocfi Since eaoh of A and 3 is an equivalence

elasa, there Is 4 sequence a in A and a sequence b in 1|

and a*b ie a Cauchy sequenoe of rational numbers; call it o,

Let Cafx |x is a Cauchy sequence of rational numbers and

x— o l Hence A*B exists.

Suppose there are tvo equivalence classes C and C*

such that k*BssC and A^BacC*; that is, there is a sequence

a in A and a sequence b in B such that a*b is in C} and

there is a seqmen.ee a* in A and a sequence b* in B suoh

that a' .b* is in C*. But a.b^a* «b*. Since a*b is in C

and a.'* to* is in 0' and s »b—a'«b*, then C«C*.

Theorem 2,12

If each of a, b, and c is a Cauchy sequence of

rational numbers, then a4(btc)x(a+b)+c.

Proof: For every positive integer n,

&2P+Ct^4-cn)3s(an*bn5+o;n. Therefore, for every integer n,

10

| {tojj.+onXl» [ ( ) •frOyil I aO«

Hence a+{b*c)=:(a.»-b)+e, It may also be stated that

a+(b*c)— (a+b)+c.

Theorem 2,13

If oaoh of A, 3« and C Is an equivalence olaae, then

A+(3+C)=(A«.B)+0.

Proof j mere 1® an & In A, b In B, and e in 0,

A* (B*0)srXas|x is * Cauchy sequence of rational numbers

and x —a+(b+c)^. (A+B)+C=:3fes£y jy is a Cauchy sequence of

rational number a and y^(a*b)*c3. Since (a+b)+c is in Y

and a+(b+c) is in X and (a+b)+c^a+(b*o), then X=X.

Theorem 2.14

If each of a, b, and e is a C&uohy sequence of rational

numbers, then a»(b»c)={a»b).o.

Proof: For every positive integer n,

% * * % • Therefore, for every positive

integer n, I6» ~0, Hence

a»(b*c)=(a*b)*0. It may also be stated that a*(b»o)—(a*b)•«,

Theorem 2.15

If each of A, 8, and C is a i equivalence class, then

A*(B«C)=(A*3)*C.

Proof: There is an a in A, b in B, and o in C.

A*{3*0}sXxfx |x is a Cauchy sequence of rational numbers

and x—a*(b*c)^. (A* 3) • CwXsr y Jy is a Cauchy sequence of

rational numbers and y^(a«b)»c]f, Since a. (b*c) is in X

11

and (a»b)*o Is in X and a»(b»o)2i (a*b)»o, than X*Xi that

ia, A* (B«C )a(A *B)*C«

Theorem 2,16

If each of a and b 1b a Cauchy sequence of rational

numbers, then a+bsb+a*

Proofs For every positive integer n,

Iienoe, for ever/ positive integer n,

| (a^bjjJ-Cb^aa) |*0, Therefore, a*b=b*a. It also say

be stated that a+b££b+a.

Theorem 2,17

If eaoh of A and B is an equivalence class, then

A+BsrB+A.

Proof: There is an a in A and b in 8. A+BecXs

?x|x la a Cauchy sequence of rational numbers and x^a*bT*

3+A=X=£y|y ia a Cauchy sequence of rational numbera and

y b+a^. Since a+b ia in X and b+a is in I and a+b^b+a,

then XaY.

Theorem 2.18

If each of a end b Is a Cauchy sequence of rational

numbers, then a»b=b*a.

Proof; For every positive integer n, ajj.b^bn'ajj.

Therefore, for every positive Integer n, I | * © »

Hence, a*b=rb«a. It also may be stated that a*b~b«a.

Theorem 2.19

If each of A and B la an equivalence class, then

A*BsrB*A.

12

Proofi There Is an a 1m A and b in B. A*B»Xss£x|x: is

a Cauchy sequence of rational numbers and x-a-bj.

B*A=X»£jr|y is a Cauchy sequence ©f rational numbers and

y^b»aj. Sine© &*b it in X and b*a is in X and a«b^b«a,

then X»Y.

Theorem 2.20

If each of a, b, and c is a Cauohy sequence of rational

nuiaber®, then a»(W@)*(&,l>)4'(re)»

proofs For every positive integer n,

| aa»(bB*eIl)-(aB'b^^aB»0ll)|*0. Hence, a«Cb4@Ma*b)4{a*c)*

It also may be stated that a* (b+o)^(a*b)«.(a*c).

Theorem 2,21

If each of A, S, and C it an equivalence class, then

A* (B4,C)«S(A»B)+( A»C) *

Proof: There is an a in A, b in B, and c in C.

A*(BtC)sXsfx\x is a Cauchy sequence of rational numbers

and (A*B)+(A«C ) ssY=£y|y is a Cauchy sequence

of rational numbers and y^(a*b)+(a«c)^. Since a*(b+c)

is in X and (a»b)«f(a*c) is in Y and a«(b«-o)^(a*bMa*o),

then Xsl. lot® that rt*rt tale® preference over % M as in

regular arithmetic.

theorem 2.22

There exists a unique equivalence class Z such that

if A is an equivalence class, then A+ZaA.

13

Proofs Let e be a positive rational musber. L®t

z*£o, 0* 0, . « ,y* L®t Zsfmjx la a Cauahy sequence of

rational numbers and x-zj. Let 1 be an equivalence olass;

there is an a in A, and there la an x in %# fhere Is a

poaitive integer 1 aueh that for integers n>H, )xn-o|-£e.

Then, !(%•%}-%)» lx^-C^-%) I* l%~® I Therefore,

a+x^ Hence there la a Z auoh that A*Z»A«

Suppose there is a olae® A ant two clauses Z and Z*,

auoh that A+ZsA and A+Z'«cA# There is an a in A» s in Z,

mud a* in z*# Then a#»a^a and a*g*^ a, Therefore

&+z- a+z*. Hence there la a positive integer S such that

for Integer® a>», ^ a ^ ^ M a ^ s ^ ) | c e. men

I <<#, Henoe,

a^n*. Sine® U s la z and z* la in Z' and *', then

z*z*. fheoreia 2,23

If A la an equivalence olaaa, then there is a unique

equivalence class A* such that A+A'aZ.

Prooft Let e he a positive rational number#

**fo, 0, 0, , . ,J 18 in Z* there is an a in A, Let

a1 he the sequence such that for every positive Integer n,

a*n®"®»' Obviously, a1 1® Cauchy. Let A'afx |x ia a Cauohy

sequence of rational numbers and x ^ a'J, For every positive

integer n, ICa^a'^)^ H{an-an)-o|«0^e. Henoe a*a*— s«

Therefore, there la a class A' auch that A*A'*Zf

14

Suppose there are two Glasses, A' and A", such that

A*A'sZ and A4»AW«Z, There la an & in A, a* in A1, and

a8 in A"; then a+a1CLz and a*aM^z. Hence a+a'^ a+a",

Ilenoe, there is a positive integer M such that for integers

n >I, |{an*& * nMa n+aMn) I e. Therefor®, for integer®

n>H,

Therefor® a'^ aM. Sine© a' i§ in A' and a" is in am and

a'— a", then A*»AM»

Theorem 2.24

There exists a unique equivalence class 0 such that

if A ia an equivalence class, then A»U*A»

Proof: Let e be a potitlve rational number. Let

u»£l, 1, 1, • » * Let Ust c |x Is a d&uehy sequence of

rational numbers and x ^ u j If A is an equivalence class,

then there is an a in A and an x in U. There is a positive

rational number d such that for every positive integer n,

Ja^l^d, There is a positive integer N such that for

integers n>K, |xQ»ll<|. Then for integers n >H,

I e /Xji-ajJss/ajJ* |xa«ll«c d»|*e. Hence, a*x — a. Therefore,

there is a class U such that A*lfeA.

Suppose there *re two classes, U and U', such that

if A is any equivalence class, then A»UsA and A'U'aA.

Let ax 2 $ 2 9 f 4 and Aafx |x is a Cauchy sequence of

rational numbers and x^aj; then a is in A. There is a

u In U and u' in U*. Then a«u—a and a«u'^a. Henoe

15

a»u —a*u'. Now since e Is a positive rational nuaber,

there is a positive Integer N such that for Integers

n>N, Igl I • 2e« Hence,

| %-n'n 1^ ». Hence, u^u*» Since u is in 0 and u1 is

In U* and u^u', then U»U',

Theorem 2*25

If each of x and y is a Cauohy sequence of rational

numberb and x^y, then there is a positive rational nuaber

d and a positive integer S such that for every integer

n>N, |xn-yn| >d.

Proofs Since xjky, there is a positive rational

nuaber 3d such that for every positive Integer I there

is an Integer n>M such that |xn-yn| "Z. 34. Since x is a

Cauchy sequence, there is a positive integer Sj__ such that

for integers a,n>N^, then ) I ^ l a a positive

integer N2 such that for integers n>]*2i then I ya«*ym |«£d»

Let Komaxlaum Ig^. There is an integer p >N such

that |xp-yp| > 3d and for any Integer n>K, ljn*lpl<:<1

and |xn-xp| * d. Mow either xp-yp>3d or xp-yp<-34.

If xp-yp£3d, then x^yns(xn«Xp)+(xp-yp)• (yp~yn) >-d+M-d^d.

If Xp-yp -3d» then ytt-*n*(yn«yp)+(^p~xp)+(*p-

xn) > -d+3d-d*d.

Hence, for Integers n>N, |%~ya|>d«

Theorem 2,26

If A Is an equivalence class and A 1® not Z, then there

is a unique equivalence ©lass A* such that A*A*»U.

Proof: Let a be a positive rational number.

u»£li 1, 1, . » is In 0, There la an a In A* Sine®

a^z, there Is a positive rational number d and a positive

Integer auch that for every integer n > N 1 # |an|> d.

Let b be the sequence suoh that b^wd If & - ail& ^nIsaJi

if n > N 1 . Clearly, b is a Cauchy sequence and b— a and

) bn| Z d for all n. Since b jlQ for every a, th* sequence

a* where a»nasJL is a sequence of rational numbers• There

1® a positive lutes®** % 0Ua3a that' f o r integer®

V b J ^ - a 2 - **«, » -j^-j •

Henoe, for integers m,n>N2>

a#«-a* Is lt» «• i » n« b^

I

b -b I. 1 .. °n "» I |T^a*% )

< e • d 2 * ^

3 C«

Hence a* is a Cauchy sequence, Clearly, b*a*^u» Since

b a, a*a«-^u» Let A#ar$x)x is a Cauehy sequence of rational

numbers and x— a*^» Hence A*A«atJ«

Suppose there are two classes A* a M A#® such that

A*A*»U and A*A*»=U. There is an a in A, a* in A*, and

a»* in A##, Then U*@.*—VL and »•&**—u# Hence a»a*^8,«a®^#

There is a positive integer such that for integers n>Nj,

IT

| J*: Let Nwnaxiraum F o r n >K,

|%•a*n»att«a,»*n| e*d and also )&n| >d. Then

U n | * |a*n-a**n|* |an*a*n-an»a**n| c ®*d, Sinoe \&n | >d t

|a*n-a*»a|< <1 a e. Hence, a*^a»*. Since a*

is in A* and a»* it in A** and a.*4t a**, then A**A»«,

Theorem 2.27

U»*U,

Proof: Let u 1 , 1# • • »*J • u is in U« There

is a u* in U*. Sinoe U*U*sU» u. Let e be a posltlre

rational number. There is a positive integer U such that

for integer® n>N, Ju^ • j e. Then

I V* u*n I * l u V * % I® | 1 * u #fT% |s I % # u * » ' % )<•• Hence

u—u*, Since u is in U and u« is in U* and u^u», then

IfeB*.

CHAPTER III

ORDER AMD COMPLETENESS OF THE CLASSES

Definition 3,1

If x It a Cftuehy sequence of rational numbers, then

x is sailed positive if and only if there Is & positive

rational number d and a positive integer N such that for

integers n >N, x^^d#

Definition 3«2

X is in set P if and only if X is an equivalence

©lass and X contains a positive Gauchy sequence.

Theorem 3.1

Bet P exists and is a subset of the set of equivalence

classes*

Proof j Clearly 1, 1, « « .j* Is a positive

sequence. Hence, Ua^xjx Is a Cauohy sequence of rational

numbers and x — i s in set P. Therefore, set P exist®

and is a set of equivalence classes*

Theorem 3.2

If A Is in set ft then every sequence in A Is positive.

Proofj Since A 1® in set P, there is a sequence x

in A such that x is positive. Suppose i is In AJ then

a~x. Since x is positive, there is a positive rational

number 2d and a positive integer such that for integers

18

19

a >N^, xQ>2d. There Is a positive integer Mg such that

for Integers n > N2» Jx^ajj) <: d, Let Msm&xlmm IgT*

Then for integers n >N, ^ > 2 ^ and )xn-anj d. Hence,

ajj>xn-d >2d-dxd. Therefore, a Is positive.

Theorem 3 . 3

If eech of A and 9 is an equivalence clas® in set P,

then A+B is in set P.

Proofs There is an a la A and b in B. Both a and

b are positive, There 1® a positive integer M|_ and a

positive rational number d^ such that for integers n >8^,

an>t5l* There is a positive integer Hg and a positive

rational number d2 such that for integers n >H2, b n>d 2.

Let Nfifflaxlmun and |aaaini!aum for

§J|

*|«d, Heno© 1® positive.

Therefore A+B is in set P.

Theorem 3.4

If each of A and B Is an equivalence el&s© in set Tf

then A*B is in set P.

Proofs There is an a in A and b in B. Both a and

b are positive, There is & positive integer and a

positive rational number d^ such that for integers n >SJ »

^n^l* There 1« a positive integer Mg and a positive

rational number d2 such that for Integers n >N2, b n>d 2.

Let M«maximua M%1 and ds(minimum $11# dg|)8. Then

20

for Integers n>N, >d« Herxoe, a*b 3.8

positive, Therefore 4*3 is la set P,

Theorem 3*5

If A is an equivalence class, then exactly oil® of

the following holds: A ie in set P, AnZ, A* ia in net P»

Proofs There is an a in A, First prove that at

most one holds. Suppose both A is in set P and A»2,

This implies that a is positive and a ^ z which Is impossible,

3upposo both A* is in set P and A=2. This implies that

a^aj but since z~z, a+z£z+z2lz which implies that z

ia in A' which oontradiota the supposition that A' was in

set P. Suppose both A is in set P and A* la in eet P.

This implies that A+A' is in set P; but A+A'sZ, This

inplies that 2 la in set P which is impossible,

low prove that at least one holds. Suppose AjiZ,

then there is an a in A so that a^x. Hence there Is a

positive integer and a positive rational number 2d

such that for lntesere n >1^, )an | >2d» Since a la a

Cauchy sequence, there la a positive integer Ifg such

that for Integers m,n>JS2» Hsasaxiausi

N2"|. Let p be a positive integer such that p>8,

then |ap| >2d; and if n is a positive integer such that

n >N, then |an-ap)<^ d. Since jap) > 2d, either a p>2d

or ap <~2d. If a p>2d, then ans:(axl-ap)+ap> -d<fr2ds=d.

Thus, if ap> 2d, then a is positive and A is in set P.

21

If a p^-2d, then an=(&n-ap)+ap<d-2d»-d whloh implies

that a* Is positive. Hence, If e,p^-2d, then a* is positive

and A* Is in aet P.

Definition 3.3

If ®aoh of A and B is an equivalence class, then

A-^B if and only if B*A* is in aet P,

Definition 3.4

If ©ash of A and B is an equival®nee ©lass, than

A ^ B meant either A ^ B or A»B.

Definition 3.5

If a set of equivalence classes, then A is as

upper bound of d if and only if A is an equivalence elass

and for every X in d, X -£A.

Definition 3.6

If d i ® & set of equivalence classes, then A is a

least upper bound of d if and only if A la an upper bound

of d and for every X which la an upper bound of d, A^X.

Theore* 3.6

If d i# a noa*®apty set of equivalence classes with

upper bound 3, then has a least upper bound T.

Proof: Suppose d la a non-empty set of equivalence

classes and B is an upper bound of d* if there exists

a class Ai in d such that for every A in d, A-£A^, then

Aj is the least upper bound of d. If thia is not true,

then if Ag la in d, there is an Aj in d auch that Ag Aj.

22

Lot A "be In ck and a be in A, Let b be in B. There

exists a positive Integer 0- such that, for Integers m,n > G,

Let H be the least Integer such, that

Let c be the sequence such that for every positive integer

n, on=H. c is obviously Cauohy. Let C be the equivalence

class containing c« For every integer n > 3,

cn~^®H~bn-b^:j*bG^

5--(H-'b0^1)-(bn-b(^1)

2 1

»-«

Hence c+b* is positive, Henoe B«£0. Therefore C ie an

upper bound of o(.

•There exist a positive integer and a positive

rational number d such that for Integers n>H^» ^^*

There exists a positive integer X2 «Mlh that for integers

m,n >N2, 1©®**%!^ |* there exists a positive Integer R^

such that for integers rn,n>Nj, 1%-a^J^I* Let !?*aaxlfflt»

K ' *2' "jl*1- Let M be a positive Integer such that

M >3(0^-8^). Let f be tha sequence such that for every

positive integer n, f^ao^M, f is obviously Cauohy. Let

F be the equivalence olasa containing f. If n>N, then

an- (0jj-M)

s V c n + I

»(an-aN)*(aK-cN)+(oN-on)*M

23

> *|*|* C ) •?( eg-a,j|)

jg-d+2 { Cg|-&£ }

>-d*2d

sd*

Hence a*f * Is positive* Hence F^A, and P is not an

upper bound of o(, Therefore there exiata a positive integer

M such that F ia not an upper bound of Let be the

greatest non-negative integer such that the class

containing the sequence la which each tersa is H-M^, is

an upper bound of Bene©» Kg, the olasa containing the

sequence in whioh eaoh term is H-C&j+l)» is not an upper

bound of ck.

Let Let T0 to# the class containing the

sequence in which eaoh term ia t0* tQ is an upper bound

of c<. Let % be the class containing the sequence in

which each term la Q© is not an upper bound of ot.

Sow d@fin« inductively the sequence t such that if

n Is ft positive integer, then 1

tn~i"|1K if the class containing the sequence

in which each term is a n

upper bound of

tjj. if the class containing the sequence

in which each tern is tn.l"!®* is not

^ an upper bound of o{. for every positive integer n, let Tn b® the class containing

24

the sequence In which each term is tQ. For every positive

integer a, let he the class containing the ®®queao©

in which ©acii terra is tn-JL, Per every positive Integer

n, fm Is m upper hound of oL and is not an upper hound

Of o(.

Let e be a positive rational number. Let I he a

positive integer such that ii' m>I # then

tjj-tia >0 and tjj»tm<^jf, and | t R - . t m | I f n>I, then

1 % " ^ Hence If m,n>Nf then jt -t lsr |

<C jfcgj-%)* tjj-tjjj J|> • Jy * ij| < *ienoe t is & Oauchy

sequence of rational numbers. Let f he the class contain-

ing t.

Suppose T 1® not an upper 'bound of ck} that is, suppose

there exists a elans X in <K such that T«£X. Let y be a

sequence in 1* there exist a positive integer and a

positive rational mrtxr £ »uoh that for lattg.r. « > % .

©sere exlsta a positive integer 1 2 such that

for Integer# a tn>M 2, l v * k J ^ § * S*®axi®uii

then for Integer® n>», 7 n * t y B ( ? & - * * ) + ( > | p * | * 4.

Hence y+tjjr Is a positive sequence, therefore

But this contraoicta the fact that for ©very positive

integer n, fB Is an upper bound of Hence ¥ is an

upper 'bound of

Suppose there exists a class S such that S<T and 3 in an

upper bound of <K, Let a be in S, Then there exist

25

a positive Integer and a positive rational nusber &

such that for Integer® n >Slf tn~»n>d. There exists a

positive Integer Ng *uch that |# There exists a

St

positive Integer such that for integers m,n >8ij,

| tgj-tjjl Let M*aaximu» lg» f?^ • X, Then for

Integer® a >8, *V" ~ Bn ® * (tn-sn) - J|

> ' 3 * * * 2 *

> m 3 * d ~ zh ^ mt fS 4s d ** ^

3 3 a I » 3#

Henoe S<Qg« But % 1® not an upper hound of «(. Henoe

3 Is not an upper bound of

Emm T Is the least upper hound of

CHAPTER 1?

UNCOUSTABILXTX OF THE CLASSES

Theorem 4,1

There Is a subset of the set of ©quivalence classes

that 1® isomorphic to the set of rational numbers.

Proofs For each rational number r, let r be the

sequence r, r, » * .J • Obviously, each sequence r

is a Cauchy sequence of rational number®, For each

rational nu-aber r» fort an equivalence class S r such

that Hr»^x|x is a Cauchy sequence of rational numbers

and x^-r?, Let 1 be the set of all ftp defined above.

It is saslly seen that a one-to-on® correspondence

exists between the rational numbers and the elements of

Ft, It can be easily shown that the aate of the sua of

two rational numbers is equal to the sum of the mates

of the two rationale. Also the aate of the product of

two rational numbers is equal to the product of the mates

of the two rational®. Hence an Isomorphism is established

between R and the set of rational numbers.

Definition 4.1

Let 3 be the set such that a is in S if and only if

a is e sequence of Integers such that for every positive

integer n, 0 £ a n < 9 and such that If p la a positive

U

27

integer, there la a positive Integer J>p se that

Let a te as element of S, Let to be a sequence suoh that

The set of all such sequences b will be called

the set of non-terminating decimals.

Theorem 4,2

A non-terminating decimal is monotone non-decreasing.

Proof? Let a be a sequence in 3. Let to be the m n

non-terminating decimal such that bn= L#t k be

a positive integer. For argr positive integer J,

lene® b^^^ >b^ and b^-b^, >0, Hence b is monotone

non-decreasing *

Theorem 4,3

A non-terminating decimal is a Gauchy sequence of

rational numbers.

Proof: Let b be a non*terminating decimal. Clearly,

b is a sequence of rational numbers.

Let e be a positive rational number, fhere is a

2 1 positive integer k such that pp<^e. low b^^b^ * &k*l

10*4i " ~K " 10'

integer. Then

% ft and % 4 'Lfc+l < bjf 4. r d g p Let r be a positive

^k+r s bk * XT |'p|| «

. r~~ 10

^ ^ v3r*s

* "k + ( 53S+T * 1 5 ^ ? • • • • • jpjf '

- • 10*( + isfci • • • • + lo^w )

1 5 ^ * • 10 *

1- . X0

X s b k + 5 T a S F

T, , XL \ * £©£•

# H«no« 0£bj^ r - + j|f) - \ »

Hwoo# 1 ^ ^ » fe^|

Lot @ Is® a positive Integ«r, then slaoe to is monotone

non-decreasing,

^k+r " ^k+e * ^^k+r ~ ~

^ 4 ^^4® " **k)

^ 1 * X

W

x©1

and. also

bk+s " ^k+r 35 " ^ • r ~ V

(^k+r ** k + (bk+r "" k^

29

^ JL» + 1

IS1 * IP - 2 * £5*

C e.

Hence - +@| and therefor© b is Cauchy.

Theorem 4,4

If each of a and c Is in set 3 and there Is a positive

integer i so that a^c^ and If for evexy positive Integer n,

bn* 11,1(1 ®n* t,h<9n

Proof: Let k be the smallest positive integer so

that a^Oig, Without loss of generality we may assume

There la a positive integer r suoh that a^so^+r.

fher© exists a positive integer h>k suoh that a jlO,

Let w b® tli© smallest aueh positive integer» Ihen if j

is a positive integer,

a iis£+ Jjsi $

%

^ £ i P ^ 15*

" 7^* • • S 15$ • "~X 4- JEI T f, T % 10 *>--1 1G F 10*

JL Orj ^ ®»j " £ # • fa, xop

* &

" I 9'( I5KX • jpb? * . . . * )

30

%- 1 ^ 5* ©« t o w To2** * I # + 9 < _ i | _

w - V C p

A 3-£ . It? S3*"

Then

"»•: " ®*+j > ( ll* • £i l§> • I P ) " ( £ r 53& + 15* '

- Io* * 15® " lo*

r-1 i%

m w * w ^ L

~ 10v#

Hence - Sw*j|£|*p} therefore b^g»

Let fte^U Is an equivalence class and & contain*

a non-terminating decimal?* Sine© no two non-terainating

decimals are equivalent, then each class in F contain®

one and only one non-terminating decimal*

Theorem 4,5

The set F defined above cannot be put in a one-to-one

correspondence with the.sftt of positive integers.

Proofi Suppose that there is a one-to-one

correspondence between the elements of F and the set

of positive integers* Let be the element of F that

is mated with the positive integer n* The claas contains

on# and only one non-terainating decimal• Denote It

by no. There exists a sequence Qa of S so that

n % 9 Heil0# n is mated by thee® relations to I

31

i|0 and na. Let gj^l if ^ = 9 and s ^ a ^ l if ^ ^ 9 *

Sine© no term of g is 0, g ia an eleaent of S» Let to

be the non-terminating decimal formed with the sequence g»

There ia an A in P which oont&ina to. There is a positive

integer k so that k is aated with this A. This A contains

jj-Q. Since # then b , Thl & is a contradiction

and there i® no mating of the positive integers with the

elesents of F. Hence a subset of the set of equivalence

classes is uncountable.

The set of all equivalence classes has been shown

to be an uncountable, ordered, complete field#

BIBLIOGRAPHY

WcShane, Edward Jamee and Truman Arthur Botta, Real , Princeton, Inc #i 1959»

jfoflinsla. Princeton, N. J,, D* fan No strand Company,

Stoll, Robert R.f XnlFQ&wttm to 8ft Theory and Logic. San Francisco, W» H. Freeman and Company, 1963.

Suppee, Patrick, Set ffeega, frlnooton, ». J., D, Van Nostrand Company, Inc., 1961*

32