AN ABSTRACT OF THE THESIS OF
NANCY MANG -ZE HUANG
for the
MASTER OF SCIENCE
(Name)
in MATHEMATICS
(Major)
(Degree)
presented on
April 26, 1968
(Date)
Title: A CONSTRUCTION OF THE REAL NUMBERS USING
NESTED CLOSED INTERV LS
Abstract approved:
Dr.
.1-1.
Arnol
It is well- knowhthat a real number can be defined as an equiva-
lence class of fundamental rational sequences. In fact, it is also pos-
sible to define a real number as an equivalence class of sequences of
nested closed rational intervals. This paper is devoted to the latter
case.
A
Construction of the Real Numbers Using
Nested Closed Intervals
by
Nancy Mang-ze Huang
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master
of Science
June 1968
APPROVED:
Professor of, athematics
in charge of major
Head of
o Department
p artment of Mathematics
Dean of Graduate School
Date thesis is presented
Typed by Clover Redfern for
April 26, 1968
Nancy Mang -ze Huang
ACKNOWLEDGMENT
Although it is difficult to find words to completely
express my gratefulness to Dr. B.H. Arnold for his
fatherly care and patient help during the past two years,
I
must at least try to let him know how much I appreci-
ate his efforts to guide and counsel me in a manner which
has helped me to fit into this American society. Thank
you so much, Dr. Arnold.
Sincerely,
Nancy M. Huang
TABLE OF CONTENTS
Chapter
Page
INTRODUCTION
1
THE EQUIVALENCE RELATION IN F
3
III.
ARITHMETIC OPERATIONS AND ORDER IN R
7
IV.
COMPLETENESS OF R
20
BIBLIOGRAPHY
29
I.
II.
A CONSTRUCTION OF THE REAL NUMBERS USING
NESTED CLOSED INTERVALS
I.
INTRODUCTION
Cohen and Ehrlich [1] have given a development of the real num-
ber system from Peano's postulates. The step from the rational
numbers to the real numbers is made by equivalence classes of
Cauchy sequences of rational numbers.
In this paper, we assume the
development of the rational numbers as given in
[1]
and we use their
notation, then we define a real number as an equivalence class of se-
quences of nested closed rational intervals and prove that our develop-
ment is equivalent to that of Cohen and Ehrlich.
In [1], an
integer is defined as an equivalence class of ordered
pairs of natural numbers, and
a
rational number is defined as an
equivalence class of ordered pairs of integers. In our construction
of the
real numbers we again begin with an equivalence relation.
In
this paper, an equivalence relation will be defined in the set of all sequences of nested closed rational intervals.
A
real number will be
defined to be an equivalence class under this equivalence relation.
By
suitable addition, multiplication, and order, the set
of
R
all
real numbers will be made into an ordered field which will be an extension of the ordered field
in
R
Q
of
all rational numbers. The order
will have no gaps in the sense of Definition
3. 9
of
[
1]
.
2
Equivalently, every fundamental sequence of real numbers will have a
limit in
R,
false in
Q
i. e.
the converse of Theorem
by Theorem
3. 24
of [1]
)
3. 22
will hold in
of [1]
R.
(which is
3
THE EQUIVALENCE RELATION IN F
II.
Before we define our equivalence relation in the rational num-
bers, we need the concept of
nested closed rational in-
a sequence of
tervals.
Definition
We define
1:
{[a
n' b])°°1
n
as a sequence of nested
closed rational intervals if
(1) an
and
n
b
are rational numbers.
n
for all n in
(2) an < bn
for all
(3) [an
bn]
+1, bn +1] C [an,
L(a -b
(4)
n
We shall
n
)
=
0,
N.
where
in
n
N.
Lim
stands for
L
n
write
{[an, b n ]
for
}
{[a
,
n
b
00
,
n }°°
n =1
]
and use
F
to denote the set of all sequences of nested closed rational intervals.
Actually, from the definition of
and
(b
n
{[an, Ian)),
n
we can prove that
(an)
are fundamental rational sequences.
)
Theorem
If
2:
{
{a
n
,
b
n
] }
E
then
F,
(an)
and
(b
n
are
)
fundamental rational sequences.
Proof:
{fan, b
s > 0
all
in
n
Q,
n > n(c)
] }
E
F
means that
there exists
in
N.
n(E)
L(a -b
n
in
N
n
)
= O.
Hence, for any given
such that
Ian -bn <
n
n
i
c
for
4
But
[a
i.e.
a
for all
n
m
n
in
N,
- am-<bm-<bn forallm>n
in
N.
b ]
m , bm 1C [a n , n
>
<
Therefore
a
l
m
-a n
Ibn -b
for all
n, m
Hence,
(an)
We
and
n
,
b
n
] }
{[c
,
d
n
i<
and
E
l<E
are fundamental rational sequences.
n
] }
bn -a n
Ibn -a
to define our equivalence
The relation
3:
n
<
m
l
N.
(bn)
are now ready
Theorem
{[a
in
n(c)
>
<
l
'
`
in
L(a -c )
n n
= 0
relation in F.
defined by
F
is an equivalence relation.
Proof:
i) {[a
n
b
,
n
] }
ii) {[an, bn] }
iii) {[a n , b n ] }
{[a
n
,
b
n
L(c -e
{[cn, dn]
implies
] }
{[c
{[e
n
,
}
= 0
d
n
, f ] }
n n
since
n
,
F.
imply
L(b -d )
n n
= O.
c
n
{[an, bn]
}
= O.
,
is an equivalence relation in
= 0
}
{[e f ] } imply
di)}
n n
n
L(a -c ) = 0 and
n n
{[c
and
= O.
n)
Theorem 4: If {[a n, b n ]},
if and only if
L(c -a )
n n
implies
] }
= O.
{[cn, dn]
L(a -e )
n n
n
Hence,
since
L(a -c )
n n
since
L(a -a )
n n
{[a , b ] }
n n
,
d ]}
n
E
F,
then L(a n -c n )
= 0
5
Proof:
L(a -c
If
nl, n2, n3
n
in
N
for any given
0,
) =
n
I
=
max
for all
n > nl
in
N,
<
3
for all
n > n
in
N,
cn -dn < 3
for all
n > n3
in
N.
I
,
i
<
Corollary
= O.
1:
3
I
{n1, n2, n3}
L(b -d )
n n
there exist
3
I
<_ Ibn
ibn -dn _
Hence
Q,
<
an -cn
Ian -bn
n(e)
in
> 0
such that
I
Let
e
then
+
Ian- cnI
for all n
e
Icn-dni
+
> n(e)
in
N.
Likewise, we can prove the other implication.
For
Han, ion])
n
e
F,
L(an)
n
=
if and only if
a
L(bn) =a.
Proof:
L(an)
L(b -a)
n
L(bn)
=
=
=
a
i. e.
L(a -a)
n
L(b -a +a -a)
n n n
=
But
= O.
L(b -a )
n n
+
L(b -a )
n n
L(a -a)
n
= O.
= O.
Then
Therefore
a.
Conclusion: If two sequences of nested closed rational intervals
are equivalent, then their left endpoints and right endpoints form two
6
fundamental rational sequences; furthermore, if either one of them
converges, then both of them do so, and they have the same limit.
Definition
F
5: A
real number is an equivalence class of the set
under the equivalence relation -
.
We use
note the equivalence class containing the element
We denote the set of
real numbers.
all real numbers by
R
C
{[a
b
and use
,
to de-
] }
n
{[an, bill )
n'
r),
E
F.
...
for
7
III.
ARITHMETIC OPERATIONS AND ORDER IN R
Before we define our addition and multiplication in the real
numbers
we shall show that the expressions we shall use for
R,
sum and product of
=
C
pendent of the choice of
Theorem
Han, b
{[an, b
If
6:
{[an, b
n
n
n
and n
}]
}]
E
and
g
{[an
] }
=
,
bl ]
n
C{
n dn} ]
{ [ cn,
do }]
and {[c
}
are inde-
[ cn,
n
,
E
d
l
n
]
.
}-
{[cn ,dn ]},
l
then
-
(1) {[an+cn, bn+dn] }
n n n n
(2) {[ancn, bndn] } -
{
Hann
bñ+dñ
n n n n
[añcn, bñdñ]
}
}
I
.
.
Proof:
-,
(1) By the definition of
L(c --c' )
n n
L(a +c -(a' +c' ))
n n n n
But
= O.
{[a +c , b +d ]}
n n n n
^
(2) Since
n
nnn
),
(al
{[a.' cl , bl +dl ]}
(a
n
n
al sequences, by Theorem
Q
such that
Theorem
I
3. 20 in
sequence in Q.
are,
that
a
n
[
I
and
),
I
cl
I
and
L(a -al )
n n
e
in
Hence
Q.
Q,
(c'
n
(c
= 0
are fundamental ration-
c'
n n
)
and
n1(e)
and
a
for all n
c'
<
)
there exist
3. 19 in [1],
(anal)
n n
Since
for each positive
n
and
< a
1],
(c
in
= 0
and
= 0
.
n
),
L(an -al)
n
we know
in
c'
in
by
N;
are fundamental rational
L(c -c'
n
and
n
)
n2(e)
= 0,
in
there
N
such
8
Ian -an
I
2cß
<
I
Zc'
cn -cl <
n n
2a
I
in Q
for all
in
for all n
Q
n>
>
nl(e) in
in
n2(e)
N
and
N.
Hence
la n cn-ancn I<_ a
I
for all n
in
I
max {n1(e), n2(e)}
>
and
Q
I
{[a c , b d ]
n n nn
Theorem
such that if
{
in
n
,
b
n
] }
(1) f(,11
,
t
E
=
and
C{[a
g('rl
) =
C{[a
{[
c , d
n
] }
n
n
}
c , b d
] }
+c
n
n
b +d
,
n n
n n
+
2c cl
c' )
L(ancn -a'nn
Therefore
N.
{[an cln bn dln j}
n
(2)
I
_ e
= 0
.
There are binary operations
7:
[a
}
la n-an lc'n I< a 2a
I+
-cn
n
E 71
,
f
and
g
on
R
then
Proof:
The sets
f
=
( 1)' C{[an+cn,n bn+dn]
n
n
,
n
})
{[an, bn]
}E
t, {[cn, dn]
,
TIER
I
and
g
((t'11)' C{[a c ,b d ]})I{[an,bn]}E,{[cn,dn]}Er,YIER
n n n n
are subsets
of
(R x R) x
9
(t, TO
If
some
{[an, b
{[c
},
n
the pair
((t,i), 0
((t3 1))
E
{
[an, bn] }
'
E
n
n
where
f
[a' , b'
and
previous Theorem
By the
n
{ [ cn,
n
{ [ cn,
dn] }
operation on
(t,i
If
for some
)
E
then
R x R,
hence the pair
((e,i),
0
((t, i ), ,t)Eg,
then
{[c' , d' ] } E 1.
n n
By the
{[a
{[a' c' , b' d' ] }
nn nn
of
,
{[a
n
,
b d ]}
n n
R x R
We
If
dñ]
}
and
] }
5 _ ÿ'
.
and hence a binary
R,
R.
{[c
c
into
R x R
] },
n n
F,
E
it follows that
6,
is a mapping of
for
an])
n n
,
where
]}
n
nnn
f
C{[c
{[a +c , b +d ]}
n n n n
Since
{[an +cn, bn +dn] } a {[al +cn, b' +di
Thus
=
n' n
C{[a'+c',b'+d'
=
] }
b ]}' 1
C{[ a +c , b +d ]}
n n n n
=
n
{
= C{[a
F.
in
d ]}
,
then
f,
t
then
R x R,
E
b
n
into
R,
E,
n
E
, d
n
g,
' = C
{[an,
] }
t
b])
n
E
in
F. Since
where
=
{[a'c',b'd']}
n n n n
and
{[c
d ]} El
,
n
{[a c , b d ] }
n n n n
C{[a c
b d ]}
n n' n n
e
If
'
{[at,b']
}E
n
where
F,
,
previous theorem again, it follows that
and
r, =
,'
.
Thus
g
is a mapping
and hence a binary operation on R.
are now ready to define our addition and multiplication in the
real numbers R.
Definition 8: We call the binary operations
f
and
g
of
10
Theorem
write
7
addition and multiplication in
and
+R r)"
"
.R ri"
ot
for
respectively, and
R,
f(
,
and
r))
usual, we shall feel free to omit the subscript
we can prove that
R,
Theorem
)
(R, +,
9:
(R, +,
,
q ).
As
"R ".
multiplication in the real
Now that we have defined addition and
numbers
g(
is a field.
)
is a field.
Proof:
i)
Associative Laws:
{[an,
] } eE
If
{[cn, dn] } er)
,
{[en, fn] }
,
e
then
(+r))
+
=C{[a +c b +d ]}
n n n n
c{[e
(ani,
+
- C{[(añ
=
C{[a
{[a
n
,
b
n
] }
+
C{[c +e
n
,
f ]}
n n
C{[an+(cn+en),bn+(dn+fn)] }
+f
n n n
,d
]1=
+
(rl+)
,
and
(.rl)
.
=
=
C{[a
c b d ]}
nn
nn
,
C{[an(cnen),
O1'
)
C{[e n , f ]} C{[(a n c n )en , (b n d n, )f n ]}
n
bn(dnfn)]} - C{[an, bn]}
C
{[c nen,
dn n]}
11
ii) Commutative Laws:
{
If
[an, bn] }
e
{{c, dn] }
,
rl,
E
then
g + 1 =
=
C{[an,
C
{[c
C{[cn, di)}
+
n+an,
dn+bn]
_
=
C{[an+cn,
C{[
C{[
+
}
=1+ g,
bn I)
and
g
.,1
_
d
nn ,bnn
C{[a c
e
g,
c a
n n
n n
iii) Distributive Laws:
{[an, bn] }
C{ [
=
]
}
=
1
g
If
{[cn, dn] }
E
1,
{[en, fn] }
E
g,
then
(g+rl)
=
C{[a +c , b +d ]}. C{[e ,f ]} C{[(an +c n )e n ,(bn+d n)fn ]}
n n n n
n n
C{[a
=
b f+d f]}
e+c n e,
nn
n nn n n
C{[anea,
bnfn]
tity,
1R;
identity of
where
Q,
OR'
OQ
[cnen' dnfn] }
We note that
iv) Identity Elements:
the additive identity,
+ C{
C {[
and
respectively.
1
1Q
Q
1
Q
]
}
C
_
OQ, OQ
+
TI
g.
serves as
as the multiplicative iden-
are the additive and multiplicative
12
NO
{[an, b
R,
{[an-bn, bn-an]} - {[OQ,
C{[an,
of
C
{[ -b
C{[an,
bn]
n
,
{[-b, -an] }
implies
F
] }E
n
Hence
For any
Additive Inverse Elements:
F
E
L(an)
}
C{[OQ,
0
serves as the additive inverse
-C {[a
,
C{[an bn' bn-an
}
Since
# O.
n
°R
Q]}:--
b ]}
n
For any
ak
>
I
el
for some
C
{[a
b
n' n
]
}# OR'
does not have limit zero in Q,
(an)
there exists a positive element el in
I
in
] } # OR
and
}+ C{[-bn,-an]
-a n ]}
n
bn]}
we know
N,
{Lan, b
OQ]} ...
vi) Multiplicative Inverse Elements:
in
C
in
k > n
mental rational sequence, there exists
such that for every
Q
N.
is funda-
Since
(an)
N
such that
in
n1
n
e
Ian -an1l
I
ak
I
>
<
2
for all
n, m >
nl
in
for
If,
N.
k1 >
nl,
el' then
1
lanl
=
lak -(ak -an)I
1
>
lak
1
1
for all n
Hence
an
0
the same process we can prove
bn
# 0
é
=
>
n1
in
min (el, e2), ñ
N.
=
max (n1, n2),
I
then
-
lak -anl
>
el
1
el el
2
T
=
for all n
>
n1
in
N.
By
for all
>
n2
in
N.
Let
n
13
Ian
I
>
2
and
Ibn
I
>
for all n
2
>
in
n
N.
Take
at
=
b'
But for every
=
n<n,
for
1
in
e > 0
n
n<n,
for
1
=bl for
at
b'n
=
a
for
n>ñ.
n
there exist n(e)
Q,
n >ri;
n
n'(e)
and
in
N
such that
Ibn-bmI
ee_2
<
4
for all n, m
>
n(e)
for all n, m
>
n'(e)
in
N
and
Ian ^am
ee_2
I
<
4
N.
-2
e
Then
4
(<
Ia'n -a'm I- Ibbn-bm
n Ibm
I
e e
I
I
-
for n, m> ma.x(n(e), ñ)
e
22
and
2
i<_4_
b' -b' I= n-am
n m
Ian! lam e e
Ila
22
(
I
Therefore
Since
in
in
N
_ e
for
n, m >
max(n' (e), ñ).
are fundamental rational sequences.
(a')
and (b')
n
n
{[an, b
n
]}
such that
e
F,
for any
Ian -bn
I
<
e > 0
ee
4
in
there exists ñ(e)
Q,
2
for all n
>
ñ(e)
in
N.
Then
14
2
ee
Ia -b
Ian bnI -
I
Ianllbnl
<
-4e
e
-
for all n
e
>
max(ñ(e), 1)
.
2 2
Therefore L(a'n -b')
n
at
implies
< ai''1
+1
{[at,
b' }e F.
n n
<
b'
< a
< b
other hand a nn +1- n +1
for all n in N, hence
On the
= O.
b'
+l < n
< b
n
We find
2
eé
Ia n a'n
for all n
L(a na')
>
= 1.
=
Ian -bnI
IbnI
Ian
bn
max(ñ(e), ñ),
lI
E
This means
L(a na')
=
1Q.
2
2
(a
a')
n n
for any
has limit
C {[a
1,
# OR'
b ]}
i. e.
there
n' n
and (b') such that
(a')
n
n
exist fundamental rational sequences
{[a', b' ]} F and
eé
e
Z
n
Now we have proved:
4
Hence
C
{[a' b' ]}
n
serves as the
n'
multiplicative inverse,
1
,
C{[an, brill
of
C
{[a
n'
b]}.
n
This completes the proof that
We shall now define an
positive elements of R
tive sequences in
theorem.
F.
(R, +,
order relation in R.
)
First,
is a field.
we define
as the equivalence classes containing posiBefore we do that, we prove the following
15
Theorem
only if
(b
n
{[a
If
10:
,
b ]}
n
then
F,
E
is positive if and
(an)
n
is positive.
)
Proof:
is positive, i. e. for some positive
(an)
exists n(e) in
Since
{[a
n'(e)
n,
in
N
But then
bn
in
such that
-
bn - an
n
an
=
(b
n
an
for all
-> e
we know for any
bn]} e F,
n
Hence
N.
such that
N
+
e
2
>
< b
2
n
- a
+ e =
2 for all n
>
in
N.
there exists
Q,
for all n
2
By the same
is positive.
)
n
e
there
Q,
n > n(e)
-
in
e > 0
<
in
e
-> n'(e)
in
N.
max(n(e), n'(e))
process, we can
prove the other implication.
Definition 11: {[a n' bn ]} is positive in
is positive. Notation: We let
R+
=
_
(
eRI
for some
{[an, b
ER1
for some
{[an, b
Next, we will prove that if
then all
{[an' , b'
n
Theorem
positive, then
n
]}
12:
If
{[an, b
{[a',
b' ]}
n n
n
n
]}
E
,
]}
E
,
{[an, b
n
are positive
e
n
]} a
]}
{[a
b ]}
n, n
is positive.
(an)
E
is positive in
is positive in
F,
F.
in
{[a' , bi"ill
is positive.
if and only if, (an)
F
n
n
and
{[a
n
,
b ]}
n
is
1
16
Proof:
By definition, we know
(an)
is positive.
n
>
n(e)
only if
in
L(a n -a')
n
n
>
such that
N
=
a'n
-
-
an +an
max {n(e), n'(e)} in N.
is positive in
is positive if and only if
]}
-
{[an, b
e2 <
> -
Hence
in
an > e
That means for any
= O.
a'n
Then
N.
such that
N
On the other hand,
N.
exists n'(e) in
in
n
is positive means for any
(an)
there exists n(e) in
{[an, b
a -
n
n
]}
<
2 +e =2>
in
Q,
for all
Q
- {[a'n , b'n ]}
if and
in
there
e > 0
a'n
e > 0
Q,
e2 for all
0
n > n'(e)
-
for all
(a') is positive and {[a',
bñ]}
F.
Now we can say
Corollary
1:
R+
Theorem
13:
R+
=
{tE R {[an, b ]}
I
n
is positive for all {[a n ,b n ]Et }.
is a set of positive elements for
Proof:
We shall show that
(1)
+ 1
E
R+
for all
t,r1
E
R +.
r)
E
R+
for all
t,
e
R+.
(2)
t
(3)
For
t
E
R,
11
exaclty one of the following holds:
t
E
R+,
= 0,
-t
E
R+
R.
17
t, ri
If
{[an, b
(cn)
n
n
,
C {[a
(a +c
n
=
and
n)
n
(a c
n n
and only if
only if
t
E
+
and
is positive in
(an)
L(an)
=
t
0Q.
=
-t
=
is positive if and only if
Thus (3) is fulfilled, and
Theorem
2. 19
e
(an) and
3. 24 in [1], we
bd
nn ]}
,
R
E
by the Corollary of
C {[0
Q
,
0
]}
(
-bn)
R +,
g = OR,
-g
e
R+
must hold.
is a set of positive elements for
R+
if
F
if and
if and only if
R
so
,
is positive. Hence, by Theorem
(-an)
t
=
n
is positive in
]}
= OR
-a n ]} e
C {[_b
3.26 in [1], exactly one of
t
where
d ]}
Hence
Q.
C{[a c n
n
n
n
Therefore
C{[an' b ] }'
n' n
{[an, b
Q.
n'
By
=
if an only if
R+
C {[c
=
exercise
By the
Q.
that (1) and (2) are fulfilled. If
12,
bn ] }' q
are positive in
)
C{[an+cn,, b
e
+c
+d ]}
n n n n
Theorem
n'
are positive in F.
d ]}
are positive sequences in
know
g +
{[c
and
]}
t=
then
R +,
e
and Definition 3.
5
R.
in [1], we have the follow-
ing theorems.
Theorem
tion in
14:
The set
T
=
{(t, l) Irl -; e R +}
is an order rela-
R.
Notation: We write
lit
ually, we omit the subscript
Theorem
15:
(R, +,
<R
nu
(iii >R
"
if
R.
,
<
)
is an ordered field.
(t,ri) e T.
Us-
18
So
far we have defined addition, multiplication and an order re-
lation in the real numbers
and we have shown that (R, +,
R,
is
<)
,
an ordered field. Now, we are going to prove the ordered field
R
of real numbers which we made is an extension of the ordered field
Q
rational numbers.
of
Theorem
E(a)
of
E
is an isomorphism of
C {[a, a]}
=
The mapping
16:
into
Q
into
Q
such that
R
preserving
R
addition, multiplication and order.
Proof:
is a one to one mapping of
E
C {[a, a]}
=
only if
L(a -b)
C {[b, b]}
a, b
If
E
= 0
Q,
into
Q
if and only if
{[a,
if and only if
a
=
all -
R.
For
in
{[b, b]}
if and
F
b.
then
E(a+b)
=
C{[a+b, a+b]}
E(ab)
=
C{[ab,
-
C{[a, a]}
+
C{[b, b]}
=
+
E(a)
E(b),
and
Thus,
Q
E
ab]} - C{[a, b]}
E(a)
C{[a, b]} -
E(b)
preserves addition and multiplication. Also,
if and only if b - a
sequence in
if and only if
Q.
C
> 0
in
so that
Q,
On the other hand,
{[b b]} - C {[a a]} >
(b -a)
a
in
R,
in
< b
is a positive
C {[a a]} < C {[b b]}
0
.
so that
C
it
R
{[b-a, b-all
19
is a positive element in R,
Q.
and
Thus
E
a< b
in
i. e.
if and only if
Q
is a positive sequence in
(b -a)
C
{[a a]} < C {[b b]}
in
R,
preserves order.
We have
just seen that the ordered field
bers is isomorphic to
we will identify
Q
the symbols
and
a
a subfield of the
with its image in
C
a, a ]}
.
Q
of
real numbers
R
rational numR.
As usual,
and use interchangeably
20
In this paper,
IV.
COMPLETENESS OF R
first
we have defined
real numbers as equivalence
classes of sequences of nested closed rational intervals; second we
of all
have proved that the set R
real numbers is an ordered field
which is an extension of the ordered field
Now we
are going to prove that
sequence in
results.
we need some preliminary
a
i.e. every Cauchy
is complete,
is convergent. Before proving this completeness,
R
Theorem
R
of rational numbers.
Q
For every real number
17:
rational number
such that
e
e
in
0 < e < s
in
> 0
there is
R,
R.
Proof:
Suppose
E
C{[a
=
definition, we know
tive
in
in
e'
for all
n > n(e')
Hence
C
>
is positive in
n(e') in
e'
,
P
an - e'
2
< b
-
an -
n
e'
(an)
> 0
in
for all n
2
is a positive element in
R.
R
in N.
It follows
]}
s = C
18:
by
R,
such that a n-> e'
N
that
Theorem
in
> 0
Then, for some posi-
Q.
Therefore,
N.
and
N,
2 b n- 2
e'
8
n' n
(an)
in
-
{[an-
Since
R.
there exists n(e') in
Q,
for all n
R
in
b
{[an,b.n]}
> C
{[
_
2,
l
2
]}
e'
=
e> 0.
2
is a fundamental rational sequence in
Q
21
if and only if
is a fundamental rational sequence in R.
(an
Proof:
By the
previous theorem, we know, for any real number
there exists
in
e > 0
Q
there exists n(e) in
n, m
in
-> n(e)
in
for any
Ian-am
I
in
< e
QC
is fundamental in
la n -a m
la n -am
R,
I
<
But for this e
.
E
in
< e
I
for all
E
in
in
n, m > n(e)
> 0,
for all
Q
n, m > n(e)
On the other hand,
R.
there exists n(e)
whenever
Q
0 < e <
is fundamental in
(an)
in
e > 0
such that
N
It follows
N.
Therefore
N.
such that
> 0,
E
such that
N
Hence
N.
(an)
Q.
With the aid of two theorems above, we can prove the next.
Theorem
1
9:
{[a , b ]}
n n
If
E
g
in
then
R,
L(a
n
)
=
L(b
n
)
_ e
Proof:
The proof is by contraposition.
L(an)
Suppose
#
f,
and
in R; we shall prove that L(an -a') # O. Since
]}
b'
n
n'
{[an, bn]} is a sequence of nested closed rational intervals, by The=
C {[a'
orem
(an)
2,
18, we know
since
ann-
(an)
L(an)
I
>
E
and
and
n
)
are fundamental in
(bn)
R
Q.
are fundamental in
then there exists
,
in
(b
for all large
n
E
in
>
O
N.
in
R
From Theorem
R
too.
But
such that
In other words
.
22
>
an-C{[a'n b'
,
c
}I
or
I
for all large
Q
in
n
such that
c
C{[an-bñ, an-añ]}I
for large
n
of absolute value (see page 68 of [1]
I
e>
From Theorem
N.
> e > 0
>
),
in
0
17, we
can find
e > 0
in
But by the definition
N.
it follows that
C{[an-bñ, an-añ]}I
=
C{[an-bñ, an-al',1]}
-a'n ]}I
^
- C{[a -b'n a n -a'n ]}
n
C{[ann, an_a]}I
=
C{[an-bn, an_añ]}
or
C{[a -b' , a
n
n
n
,
If
for large n
in
L(a -a'n )
n
and
# 0
N,
then
(a
{[an, b ]} .
n n
n
`5
>
e>
0
is positive, therefore
-a')
n
.
If
,b°-a
C{[a -b', a -a' ]}I= - C{[a -b' , a -al)} - C{[a'-a
n n n n
n n n n
n n n n
for large n
in
L(a'n -a n )
and
# 0
N,
(a'n -a
then
{[an, b
n
]}
j
`5
.
n
)
is positive; therefore
e>
0
23
We now get a conclusion:
then
in
{[an,
b]}
n i
then
R,
L(an)
L(an)
we know
i. é. for any
;
t
=
L(bn)
n
=
,
=
Since
b
n
{[an, b
{[an, b
n
n
]}
]} e
stronger statement that for any
;
] }e
{[an, brill
if
F,
n
n
t
E
such that
n
,
b ]}
n
e
=
L(b
,, therefore we
{[an, bn]}
n
{[a
we are able to find a
F,
t and L(an)
E
can only belong to one
{[an, bn]}
real number
,
t
E
n
)
=
t
.
have a
there exists a unique
F,
L(an)
and
=
L(bn)
= g
.
The following statements are consequences of Theorem 19.
Corollary
a
in
such that
Q
te
If
1:
for any
R,
' -a I<
I
in
E
E
> 0
in
there exists
R
R.
Proof:
If
L(an)
=
g =
,
such that
a
that
=
.
I
in
b
n' n ]}
Then, for any
an(E),
I
<
I
Corollary
2:
If
,< a <i
in
R.
E
> 0
whenever
E
then
from Theorem
R,
C {[a
e -a n
,
This completes the proof.
.
such that
t
n
t
#
But from the Corollary of Theorem 4,
.
Now we can say for any
real number
{[a
if L(an)
{[an, bn]} e F
for any
-a1<
g <
r
E
in
n
>
R
n(e)
19, we know
there exists
in
N.
In
in
n(e)
N
particular, if
is as required.
in
R,
there exists
a
in
Q
such
24
Proof:
Since
is an ordered field, by Theorem
R
there exists
e =
min
{-
T1-0
there exists a
- e<
- a <
Y
in
Corollary
,<
<
such that
Q
ri
in
R.
Let
r, e
R
and
e
in
R,
Is -aI
Therefore
R.
t<
<
E
R
i. e.
+e<11.
e< a <
l;
in
> 0
then
is Archimedean.
R
3:
t
then for this
> 0,
in
e
such that
R
,E
3. 16 of [1],
Proof:
For
t
0 <
bers such that
in
< ri
0 <
medean (see page
a
t
<
69 of
R,
< b <
let
t
and
a
+ ri
in
> b.
nt
But
>
na
Since
R.
is Archi-
Q
[1]) and the embedding isomorphism preserves
addition and order, there is an interger
na
be two rational num-
b
> b > q
in
R,
in
N
such that
therefore
R
is Archime-
n
dean.
Conclusion: From Theorem
quence
points
{[an, b
(an)
n
]}
2,
we only know that for every se-
of nested closed rational intervals, the left end-
and the right endpoints
(b)
form fundamental sen
quences. But now we know more. That is, they are convergent, and
converge to the real number which the sequence
{[an, b
n
]}
repre-
sents. We also know every real number is the limit of at least one
fundamental rational sequence. Finally, we know that for any real
25
number, we can find a rational number as near the real number as
desired;
no
matter how close together two real numbers are, there
are rational numbers in between.
are now ready to prove the
We
completeness of R.
Theorem
20:
is complete.
R
Proof:
(p) is
Suppose
then, for any
in
k
p >
-
k
nk
where
R,
there exists nk in
N,
n
is true for all
Cauchy in
k<
in
p <
N
= C {[ap,
n
bp]}
n
such that
+k
But, on the other hand,
N.
nk
k
P
=C
{[an k, bn k]}
L(ak)
n
=
nk
=
L(b nnk)
,
and
n
a
for all n in
n(k)
and
n' (k)
n
k<- nk <- b n k
Therefore, for any
N.
in
N
nk
such that
k
in
N,
there exist
'
26
nk
-a
nk
<
I
n
+
k
for n
>
for n
> n° (k)
in
n(k)
N
and
in k -a n k I< k
in
respectively;
N,
i. e.
n
nk
-
n
an(k) < an k < nk
<
for
n > n(k)
for
n
in
N
in
N.
N
and
nk
n
< b
k
< b
(k)
k
Hence, for all
p
am
m
in
<
n
max
=
n
max {an(k) - k
=
-
m
min
k
k
{nk k < m}
I
n
{bn
+
>
nB
(k)
there exist
N,
=
b
+
k< m
I
k
I
}
k < m}
in
N,
in
Q,
in
Q,
such that
a
m -1
<
<
tpp <b m-<b m -1 for all m
in
in
N.
P
But then we have
and
{[a
b ]}
m' m C
{[a
- pm
_>
b
m -l' m -
]}
1
N
and
for all m
in
N,
27
nm
bm-am
I
-<
i
I
bna (m)+
m
°
nm
(an(m)° m )
n
n
m
an(m)
I?
+b m
m n' (m)
=
1
nrn)
I
4
<-+-+=m
m
m m
2
tends to
as
0
1
1
tends to infinity. Hence
m
nested closed rational sequence. For this
there exists some
in
g
such that
R
n
a
1<
a <
m - m
m
p-
in
and
N
n
=
I
I
p
-
I
<
We have
< b
-
p > n
n
m
p-bns
L(am)
m
m
L(b
=
m)
is a
]}
we know
=
,
Then
.
for all m
in
N;
1
+
- bnm
n' (m) m
<
m
in
m
N.
m
+13
+
n
m
Hence
nm
n
n
m
m
- an(m)+ an(m)
I
1m 1m 1m 1mm
=4.
+
+
+
is Archimedean,
Since R
I
m
^5
<
fore
b
,
n
n(m)
I
m
{[an1, brnI },
-1< a < < < m m
n(m) m - m - - bm- bn' (m) + 1
anm
for all
{[a
for large
4m <
e
for some m
in
N.
There -
Hence
R
is complete.
just proved that the ordered field
R
which we have
E
p
in
N.
constructed by using nested closed rational intervals is complete and
28
Archimedean. Although the way we build up the system of real num-
bers from rational numbers is different from that of Cohen and Ehrlich, our end results (i. e. the field of real numbers) are isomorphic
by Theorem 5.
3
of [ l ] .
29
BIBLIOGRAPHY
1.
Cohen, L. W. and G. Ehrlich. The structure of the real number
system. Princeton, N. J. , Van Nostrand, 1963. 116 p.