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