Real Number System as a Complete Ordered Field and Archimedean Property - CSIR-NET Mathematical | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


2 The real numbers as a complete ordered ¯eld
In this section are presented what can be thought of as \the rules of the game:"
the axioms of the real numbers. In this work, we present these axioms as
rules without justi¯cation. There are other approaches which can be used. For
example, another standard technique is to begin with the Peano axioms|the
axioms of the natural numbers|and build up to the real numbers through
several \completions" of this system. In such a setup, our axioms are theorems.
2.1 Field Axioms
This ¯rst set of axioms are called the ¯eld axioms because any object satisfying
them is called a ¯eld. They give the algebraic properties of the real numbers.
A ¯eld is a nonempty set F along with two functions, multiplication£ :
F£F!F and addition + :F£F!F satisfying the following axioms.
3
Axiom 1 (Associative Laws). If a;b;c2 F, then (a +b)+c =a+(b+c)
and (a£b)£c =a£ (b£c).
Axiom 2 (Commutative Laws). If a;b2F, then a +b =b +a and a£b =
b£a.
Axiom 3 (Distributive Law). Ifa;b;c2F, then a£ (b +c)=a£b+a£c.
Axiom 4 (Existence of identities). There are 0; 12F such that a+0 =a
and a£1=a,8a2F.
Axiom 5 (Existence of an additive inverse). For eacha2F there is¡a2
F such that a+(¡a)=0.
Axiom 6 (Existence of a multiplicative inverse). For each a 2 Fnf0g
there is a
¡1
2F such that a£a
¡1
=1.
Although these axioms seem to contain most of the properties of the real
numbers we normally use, there are other ¯elds besides the real numbers.
Example 2.1. From elementary algebra we know that the rational numbers,Q,
are a ¯eld.
Example 2.2. LetF =f0; 1; 2g with with addition and multiplication calculated
modulo 3. It is easy to check that the ¯eld axioms are satis¯ed.
Theorem 2.1. The additive and multiplicative identities of a ¯eldF are unique.
Proof. Suppose e
1
and e
2
are both multiplicative identities inF. Then
e
1
=e
1
£e
2
=e
2
;
so the multiplicative identity is unique. The proof for the additive identity is
essentially the same.
3
The functions + and£ are often called binary operations. The standard notation of
+(a;b)=a+b and£(a;b)=a£b is used here.
Page 2


2 The real numbers as a complete ordered ¯eld
In this section are presented what can be thought of as \the rules of the game:"
the axioms of the real numbers. In this work, we present these axioms as
rules without justi¯cation. There are other approaches which can be used. For
example, another standard technique is to begin with the Peano axioms|the
axioms of the natural numbers|and build up to the real numbers through
several \completions" of this system. In such a setup, our axioms are theorems.
2.1 Field Axioms
This ¯rst set of axioms are called the ¯eld axioms because any object satisfying
them is called a ¯eld. They give the algebraic properties of the real numbers.
A ¯eld is a nonempty set F along with two functions, multiplication£ :
F£F!F and addition + :F£F!F satisfying the following axioms.
3
Axiom 1 (Associative Laws). If a;b;c2 F, then (a +b)+c =a+(b+c)
and (a£b)£c =a£ (b£c).
Axiom 2 (Commutative Laws). If a;b2F, then a +b =b +a and a£b =
b£a.
Axiom 3 (Distributive Law). Ifa;b;c2F, then a£ (b +c)=a£b+a£c.
Axiom 4 (Existence of identities). There are 0; 12F such that a+0 =a
and a£1=a,8a2F.
Axiom 5 (Existence of an additive inverse). For eacha2F there is¡a2
F such that a+(¡a)=0.
Axiom 6 (Existence of a multiplicative inverse). For each a 2 Fnf0g
there is a
¡1
2F such that a£a
¡1
=1.
Although these axioms seem to contain most of the properties of the real
numbers we normally use, there are other ¯elds besides the real numbers.
Example 2.1. From elementary algebra we know that the rational numbers,Q,
are a ¯eld.
Example 2.2. LetF =f0; 1; 2g with with addition and multiplication calculated
modulo 3. It is easy to check that the ¯eld axioms are satis¯ed.
Theorem 2.1. The additive and multiplicative identities of a ¯eldF are unique.
Proof. Suppose e
1
and e
2
are both multiplicative identities inF. Then
e
1
=e
1
£e
2
=e
2
;
so the multiplicative identity is unique. The proof for the additive identity is
essentially the same.
3
The functions + and£ are often called binary operations. The standard notation of
+(a;b)=a+b and£(a;b)=a£b is used here.
Theorem 2.2. Let F be a ¯eld. If a;b2F with b6=0, then¡a and b
¡1
are
unique.
Proof. Supposeb
1
andb
2
are both multiplicative inverses forb6= 0. Then, using
Axiom 1,
b
1
=b
1
£1=b
1
£(b£b
2
)=(b
1
£b)£b
2
=1£b
2
=b
2
:
This shows the multiplicative inverse in unique. The proof is essentially the
same for the additive inverse.
From now on we will assume the standard notations for algebra; e. g., we
will writeab instead ofa£b anda=b instead ofa£b
¡1
. There are many other
properties of ¯elds which could be proved here, but they correspond to the usual
properties of the real numbers learned in beginning algebra, so we omit them.
Problem 8. Prove that if a;b2F, whereF is a ¯eld, then (¡a)b =¡(ab)=
a(¡b).
Page 3


2 The real numbers as a complete ordered ¯eld
In this section are presented what can be thought of as \the rules of the game:"
the axioms of the real numbers. In this work, we present these axioms as
rules without justi¯cation. There are other approaches which can be used. For
example, another standard technique is to begin with the Peano axioms|the
axioms of the natural numbers|and build up to the real numbers through
several \completions" of this system. In such a setup, our axioms are theorems.
2.1 Field Axioms
This ¯rst set of axioms are called the ¯eld axioms because any object satisfying
them is called a ¯eld. They give the algebraic properties of the real numbers.
A ¯eld is a nonempty set F along with two functions, multiplication£ :
F£F!F and addition + :F£F!F satisfying the following axioms.
3
Axiom 1 (Associative Laws). If a;b;c2 F, then (a +b)+c =a+(b+c)
and (a£b)£c =a£ (b£c).
Axiom 2 (Commutative Laws). If a;b2F, then a +b =b +a and a£b =
b£a.
Axiom 3 (Distributive Law). Ifa;b;c2F, then a£ (b +c)=a£b+a£c.
Axiom 4 (Existence of identities). There are 0; 12F such that a+0 =a
and a£1=a,8a2F.
Axiom 5 (Existence of an additive inverse). For eacha2F there is¡a2
F such that a+(¡a)=0.
Axiom 6 (Existence of a multiplicative inverse). For each a 2 Fnf0g
there is a
¡1
2F such that a£a
¡1
=1.
Although these axioms seem to contain most of the properties of the real
numbers we normally use, there are other ¯elds besides the real numbers.
Example 2.1. From elementary algebra we know that the rational numbers,Q,
are a ¯eld.
Example 2.2. LetF =f0; 1; 2g with with addition and multiplication calculated
modulo 3. It is easy to check that the ¯eld axioms are satis¯ed.
Theorem 2.1. The additive and multiplicative identities of a ¯eldF are unique.
Proof. Suppose e
1
and e
2
are both multiplicative identities inF. Then
e
1
=e
1
£e
2
=e
2
;
so the multiplicative identity is unique. The proof for the additive identity is
essentially the same.
3
The functions + and£ are often called binary operations. The standard notation of
+(a;b)=a+b and£(a;b)=a£b is used here.
Theorem 2.2. Let F be a ¯eld. If a;b2F with b6=0, then¡a and b
¡1
are
unique.
Proof. Supposeb
1
andb
2
are both multiplicative inverses forb6= 0. Then, using
Axiom 1,
b
1
=b
1
£1=b
1
£(b£b
2
)=(b
1
£b)£b
2
=1£b
2
=b
2
:
This shows the multiplicative inverse in unique. The proof is essentially the
same for the additive inverse.
From now on we will assume the standard notations for algebra; e. g., we
will writeab instead ofa£b anda=b instead ofa£b
¡1
. There are many other
properties of ¯elds which could be proved here, but they correspond to the usual
properties of the real numbers learned in beginning algebra, so we omit them.
Problem 8. Prove that if a;b2F, whereF is a ¯eld, then (¡a)b =¡(ab)=
a(¡b).
2.2 Order Axiom
The axiom of this section gives us the order properties of the real numbers.
Axiom 7 (Order axiom.). There is a set P½F such that
(i) If a;b2P, then a +b;ab2P.
(ii) Ifa2F, then exactly one of the following is true: a2P,¡a2P ora=0.
Of course, theP is known as the set of positive elements ofF. Using Axiom
7(ii), we see that F is divided into three pairwise disjoint sets: P, f0g and
f¡x :x2Pg. The latter of these is the set of negative elements ofF.
De¯nition 2.1. We writea<b orb>a,ifb¡a2P. The meanings of a·b
and b¸a are now as expected.
Example 2.3. The rational numbersQ are an ordered ¯eld. This example shows
there are ordered ¯elds which are not equal toR.
Extra Credit 2. Prove there is no setP½Z
3
which makesZ
3
into an ordered
¯eld.
Following are a few standard properties of ordered ¯elds.
Theorem 2.3. a6=0 i® a
2
> 0.
Proof. ())Ifa> 0, then a
2
> 0 by Axiom 7(a). Ifa< 0, then¡a> 0by
Axiom 7(b) and above, a
2
=1a
2
=(¡1)(¡1)a
2
=(¡a)
2
> 0.
(() Since 0
2
= 0, this is obvious.
Theorem 2.4. IfF is an ordered ¯eld and a;b;c2F, then
(a) a<b () a +c<b+c,
(b) a<b^b<c =) a<c,
(c) a<b^c>0=) ac<bc,
(d) a<b^c<0=) ac>bc.
Proof. (a)a<b () b¡a2 P () (b +c)¡ (a +c)2 P () a +c<b+c.
(b) By supposition, both b¡a;c¡b2 P. Using the fact that P is closed
under addition, we see (b¡a)+(c¡b)=c¡a2P. Therefore,c>a.
(c) Sinceb¡a2P andc2P andP is closed under multiplication,c(b¡a)=
cb¡ca2P and, therefore,ac<bc.
(d) By assumption, b¡a;¡c2P. Apply part (c) and Problem 8.
Theorem 2.5 (Two out of three rule). LetF be an ordered ¯eld anda;b;c2
F.If ab =c and any two of a, b or c are positive, then so is the third.
Page 4


2 The real numbers as a complete ordered ¯eld
In this section are presented what can be thought of as \the rules of the game:"
the axioms of the real numbers. In this work, we present these axioms as
rules without justi¯cation. There are other approaches which can be used. For
example, another standard technique is to begin with the Peano axioms|the
axioms of the natural numbers|and build up to the real numbers through
several \completions" of this system. In such a setup, our axioms are theorems.
2.1 Field Axioms
This ¯rst set of axioms are called the ¯eld axioms because any object satisfying
them is called a ¯eld. They give the algebraic properties of the real numbers.
A ¯eld is a nonempty set F along with two functions, multiplication£ :
F£F!F and addition + :F£F!F satisfying the following axioms.
3
Axiom 1 (Associative Laws). If a;b;c2 F, then (a +b)+c =a+(b+c)
and (a£b)£c =a£ (b£c).
Axiom 2 (Commutative Laws). If a;b2F, then a +b =b +a and a£b =
b£a.
Axiom 3 (Distributive Law). Ifa;b;c2F, then a£ (b +c)=a£b+a£c.
Axiom 4 (Existence of identities). There are 0; 12F such that a+0 =a
and a£1=a,8a2F.
Axiom 5 (Existence of an additive inverse). For eacha2F there is¡a2
F such that a+(¡a)=0.
Axiom 6 (Existence of a multiplicative inverse). For each a 2 Fnf0g
there is a
¡1
2F such that a£a
¡1
=1.
Although these axioms seem to contain most of the properties of the real
numbers we normally use, there are other ¯elds besides the real numbers.
Example 2.1. From elementary algebra we know that the rational numbers,Q,
are a ¯eld.
Example 2.2. LetF =f0; 1; 2g with with addition and multiplication calculated
modulo 3. It is easy to check that the ¯eld axioms are satis¯ed.
Theorem 2.1. The additive and multiplicative identities of a ¯eldF are unique.
Proof. Suppose e
1
and e
2
are both multiplicative identities inF. Then
e
1
=e
1
£e
2
=e
2
;
so the multiplicative identity is unique. The proof for the additive identity is
essentially the same.
3
The functions + and£ are often called binary operations. The standard notation of
+(a;b)=a+b and£(a;b)=a£b is used here.
Theorem 2.2. Let F be a ¯eld. If a;b2F with b6=0, then¡a and b
¡1
are
unique.
Proof. Supposeb
1
andb
2
are both multiplicative inverses forb6= 0. Then, using
Axiom 1,
b
1
=b
1
£1=b
1
£(b£b
2
)=(b
1
£b)£b
2
=1£b
2
=b
2
:
This shows the multiplicative inverse in unique. The proof is essentially the
same for the additive inverse.
From now on we will assume the standard notations for algebra; e. g., we
will writeab instead ofa£b anda=b instead ofa£b
¡1
. There are many other
properties of ¯elds which could be proved here, but they correspond to the usual
properties of the real numbers learned in beginning algebra, so we omit them.
Problem 8. Prove that if a;b2F, whereF is a ¯eld, then (¡a)b =¡(ab)=
a(¡b).
2.2 Order Axiom
The axiom of this section gives us the order properties of the real numbers.
Axiom 7 (Order axiom.). There is a set P½F such that
(i) If a;b2P, then a +b;ab2P.
(ii) Ifa2F, then exactly one of the following is true: a2P,¡a2P ora=0.
Of course, theP is known as the set of positive elements ofF. Using Axiom
7(ii), we see that F is divided into three pairwise disjoint sets: P, f0g and
f¡x :x2Pg. The latter of these is the set of negative elements ofF.
De¯nition 2.1. We writea<b orb>a,ifb¡a2P. The meanings of a·b
and b¸a are now as expected.
Example 2.3. The rational numbersQ are an ordered ¯eld. This example shows
there are ordered ¯elds which are not equal toR.
Extra Credit 2. Prove there is no setP½Z
3
which makesZ
3
into an ordered
¯eld.
Following are a few standard properties of ordered ¯elds.
Theorem 2.3. a6=0 i® a
2
> 0.
Proof. ())Ifa> 0, then a
2
> 0 by Axiom 7(a). Ifa< 0, then¡a> 0by
Axiom 7(b) and above, a
2
=1a
2
=(¡1)(¡1)a
2
=(¡a)
2
> 0.
(() Since 0
2
= 0, this is obvious.
Theorem 2.4. IfF is an ordered ¯eld and a;b;c2F, then
(a) a<b () a +c<b+c,
(b) a<b^b<c =) a<c,
(c) a<b^c>0=) ac<bc,
(d) a<b^c<0=) ac>bc.
Proof. (a)a<b () b¡a2 P () (b +c)¡ (a +c)2 P () a +c<b+c.
(b) By supposition, both b¡a;c¡b2 P. Using the fact that P is closed
under addition, we see (b¡a)+(c¡b)=c¡a2P. Therefore,c>a.
(c) Sinceb¡a2P andc2P andP is closed under multiplication,c(b¡a)=
cb¡ca2P and, therefore,ac<bc.
(d) By assumption, b¡a;¡c2P. Apply part (c) and Problem 8.
Theorem 2.5 (Two out of three rule). LetF be an ordered ¯eld anda;b;c2
F.If ab =c and any two of a, b or c are positive, then so is the third.
Proof. Ifa> 0 andb> 0, then Axiom 7(a) impliesc> 0. Next, supposea>0
andc> 0. In order to force a contradiction, supposeb· 0. In this case, Axiom
7(b) shows
0·a(¡b)=¡(ab)=¡c<0;
which is impossible.
Corollary 2.6. LetF be an ordered ¯eld and a2F.Ifa>0, then a
¡1
> 0.If
a<0, then a
¡1
< 0.
Proof. The proof is Problem 9.
Problem 9. Prove Corollary 2.6.
Supposea> 0. Since 1a =a, Theorem 2.5 implies 1> 0. Applying Theorem
2.4, we see that 1 + 1> 1> 0. It's clear that by induction, we can ¯nd a copy
of N in any ordered ¯eld. Similarly, Z and Q also have unique copies in any
ordered ¯eld.
The standard notation for intervals will be used on an ordered ¯eld,F; i. e.,
(a;b)=fx2F :a<x<bg,(a;1)=fx2F :a<xg,[a;b]=fx2F :a·x·
bg, etc.
2.2.1 Metric Properties
The order axiom on a ¯eld F allows us to introduce the idea of a distance
between points inF. To do this, we begin with the following familiar de¯nition.
De¯nition 2.2. LetF be an ordered ¯eld. The absolute value function onF is
a functionj¢j :F!F de¯ned as
jxj =
(
x; x¸ 0
¡x; x< 0
:
The most important properties of the absolute value function are contained
in the following theorem.
Theorem 2.7. LetF be an ordered ¯eld. Then
(a)jxj¸0 for all x2F andjxj =0 () x=0;
(b) jxj =j¡xj for all x2F;
(c) ¡jxj·x·jxj for all x2F;
(d)jxj·y () ¡y· x· y; and,
(e) jx +yj·jxj+jyj for all x;y2F.
Proof. (a) The fact thatjxj¸ 0 for all x2F follows from Axiom 7(b). Since
0=¡0, the second part is clear.
Page 5


2 The real numbers as a complete ordered ¯eld
In this section are presented what can be thought of as \the rules of the game:"
the axioms of the real numbers. In this work, we present these axioms as
rules without justi¯cation. There are other approaches which can be used. For
example, another standard technique is to begin with the Peano axioms|the
axioms of the natural numbers|and build up to the real numbers through
several \completions" of this system. In such a setup, our axioms are theorems.
2.1 Field Axioms
This ¯rst set of axioms are called the ¯eld axioms because any object satisfying
them is called a ¯eld. They give the algebraic properties of the real numbers.
A ¯eld is a nonempty set F along with two functions, multiplication£ :
F£F!F and addition + :F£F!F satisfying the following axioms.
3
Axiom 1 (Associative Laws). If a;b;c2 F, then (a +b)+c =a+(b+c)
and (a£b)£c =a£ (b£c).
Axiom 2 (Commutative Laws). If a;b2F, then a +b =b +a and a£b =
b£a.
Axiom 3 (Distributive Law). Ifa;b;c2F, then a£ (b +c)=a£b+a£c.
Axiom 4 (Existence of identities). There are 0; 12F such that a+0 =a
and a£1=a,8a2F.
Axiom 5 (Existence of an additive inverse). For eacha2F there is¡a2
F such that a+(¡a)=0.
Axiom 6 (Existence of a multiplicative inverse). For each a 2 Fnf0g
there is a
¡1
2F such that a£a
¡1
=1.
Although these axioms seem to contain most of the properties of the real
numbers we normally use, there are other ¯elds besides the real numbers.
Example 2.1. From elementary algebra we know that the rational numbers,Q,
are a ¯eld.
Example 2.2. LetF =f0; 1; 2g with with addition and multiplication calculated
modulo 3. It is easy to check that the ¯eld axioms are satis¯ed.
Theorem 2.1. The additive and multiplicative identities of a ¯eldF are unique.
Proof. Suppose e
1
and e
2
are both multiplicative identities inF. Then
e
1
=e
1
£e
2
=e
2
;
so the multiplicative identity is unique. The proof for the additive identity is
essentially the same.
3
The functions + and£ are often called binary operations. The standard notation of
+(a;b)=a+b and£(a;b)=a£b is used here.
Theorem 2.2. Let F be a ¯eld. If a;b2F with b6=0, then¡a and b
¡1
are
unique.
Proof. Supposeb
1
andb
2
are both multiplicative inverses forb6= 0. Then, using
Axiom 1,
b
1
=b
1
£1=b
1
£(b£b
2
)=(b
1
£b)£b
2
=1£b
2
=b
2
:
This shows the multiplicative inverse in unique. The proof is essentially the
same for the additive inverse.
From now on we will assume the standard notations for algebra; e. g., we
will writeab instead ofa£b anda=b instead ofa£b
¡1
. There are many other
properties of ¯elds which could be proved here, but they correspond to the usual
properties of the real numbers learned in beginning algebra, so we omit them.
Problem 8. Prove that if a;b2F, whereF is a ¯eld, then (¡a)b =¡(ab)=
a(¡b).
2.2 Order Axiom
The axiom of this section gives us the order properties of the real numbers.
Axiom 7 (Order axiom.). There is a set P½F such that
(i) If a;b2P, then a +b;ab2P.
(ii) Ifa2F, then exactly one of the following is true: a2P,¡a2P ora=0.
Of course, theP is known as the set of positive elements ofF. Using Axiom
7(ii), we see that F is divided into three pairwise disjoint sets: P, f0g and
f¡x :x2Pg. The latter of these is the set of negative elements ofF.
De¯nition 2.1. We writea<b orb>a,ifb¡a2P. The meanings of a·b
and b¸a are now as expected.
Example 2.3. The rational numbersQ are an ordered ¯eld. This example shows
there are ordered ¯elds which are not equal toR.
Extra Credit 2. Prove there is no setP½Z
3
which makesZ
3
into an ordered
¯eld.
Following are a few standard properties of ordered ¯elds.
Theorem 2.3. a6=0 i® a
2
> 0.
Proof. ())Ifa> 0, then a
2
> 0 by Axiom 7(a). Ifa< 0, then¡a> 0by
Axiom 7(b) and above, a
2
=1a
2
=(¡1)(¡1)a
2
=(¡a)
2
> 0.
(() Since 0
2
= 0, this is obvious.
Theorem 2.4. IfF is an ordered ¯eld and a;b;c2F, then
(a) a<b () a +c<b+c,
(b) a<b^b<c =) a<c,
(c) a<b^c>0=) ac<bc,
(d) a<b^c<0=) ac>bc.
Proof. (a)a<b () b¡a2 P () (b +c)¡ (a +c)2 P () a +c<b+c.
(b) By supposition, both b¡a;c¡b2 P. Using the fact that P is closed
under addition, we see (b¡a)+(c¡b)=c¡a2P. Therefore,c>a.
(c) Sinceb¡a2P andc2P andP is closed under multiplication,c(b¡a)=
cb¡ca2P and, therefore,ac<bc.
(d) By assumption, b¡a;¡c2P. Apply part (c) and Problem 8.
Theorem 2.5 (Two out of three rule). LetF be an ordered ¯eld anda;b;c2
F.If ab =c and any two of a, b or c are positive, then so is the third.
Proof. Ifa> 0 andb> 0, then Axiom 7(a) impliesc> 0. Next, supposea>0
andc> 0. In order to force a contradiction, supposeb· 0. In this case, Axiom
7(b) shows
0·a(¡b)=¡(ab)=¡c<0;
which is impossible.
Corollary 2.6. LetF be an ordered ¯eld and a2F.Ifa>0, then a
¡1
> 0.If
a<0, then a
¡1
< 0.
Proof. The proof is Problem 9.
Problem 9. Prove Corollary 2.6.
Supposea> 0. Since 1a =a, Theorem 2.5 implies 1> 0. Applying Theorem
2.4, we see that 1 + 1> 1> 0. It's clear that by induction, we can ¯nd a copy
of N in any ordered ¯eld. Similarly, Z and Q also have unique copies in any
ordered ¯eld.
The standard notation for intervals will be used on an ordered ¯eld,F; i. e.,
(a;b)=fx2F :a<x<bg,(a;1)=fx2F :a<xg,[a;b]=fx2F :a·x·
bg, etc.
2.2.1 Metric Properties
The order axiom on a ¯eld F allows us to introduce the idea of a distance
between points inF. To do this, we begin with the following familiar de¯nition.
De¯nition 2.2. LetF be an ordered ¯eld. The absolute value function onF is
a functionj¢j :F!F de¯ned as
jxj =
(
x; x¸ 0
¡x; x< 0
:
The most important properties of the absolute value function are contained
in the following theorem.
Theorem 2.7. LetF be an ordered ¯eld. Then
(a)jxj¸0 for all x2F andjxj =0 () x=0;
(b) jxj =j¡xj for all x2F;
(c) ¡jxj·x·jxj for all x2F;
(d)jxj·y () ¡y· x· y; and,
(e) jx +yj·jxj+jyj for all x;y2F.
Proof. (a) The fact thatjxj¸ 0 for all x2F follows from Axiom 7(b). Since
0=¡0, the second part is clear.
(b) If x¸ 0, then¡x· 0 so thatj¡xj =¡(¡x)=x =jxj.Ifx< 0, then
¡x> 0 andjxj =¡x =j¡xj.
(c) If x¸ 0, then¡jxj =¡x·x =jxj.Ifx< 0, then¡jxj =¡(¡x)=x<
¡x=jxj.
(d) This is left as an exercise.
(e) Add the two sets of inequalities¡jxj·x·jxj and¡jyj·y·jyj to see
¡(jxj +jyj)·x +y·jxj+jyj. Now apply (d).
De¯nition 2.3. Let S be a set and d :S£S!R satisfy
(a) for all x;y2S, d(x;y)¸ 0 and d(x;y)=0 () x = y,
(b) for all x;y2S, d(x;y)=d(y;x), and
(c) for all x;y;z2S, d(x;z)·d(x;y)+d(y;z).
Then the function d is a metric on S.
A metric is a function which de¯nes the distance between any two points of
a set.
Example 2.4. Let S be a set and de¯ne d :S£S!S by
d(x;y)=
(
1;x6=y
0;x=y
:
It is easy to prove that d is a metric on S. This trivial metric is called the
discrete metric.
Theorem 2.8. IfF is an ordered ¯eld, then d(x;y)=jx¡yj is a metric onF.
Proof. This easily follows from various parts of Theorem 2.7
Problem 10. Provejxj·y i®¡y·x·y.