1.1 The real numbers. Supremum and Infimum of a set.

Answer:

The real number system is first of all a set {a,b,c,…} on which the operation of addition and multiplication a defined that every pair of real numbers has a unique sum and product, both real numbers,with the following properties,

1. a+b=b+a and ab=ba (commutative laws)

2. (a+b)+c=a+(b+c) (ab)c=a(bc) (associative laws)

3. a(b+c)=ab+ac (distributive law)

4. There are distinct real numbers 0 and 1 such that a+0=a and a1=a for all a

5. For each (a) there is a real number –a such that a+(-a)=0 and if a 0 there is a real number such that a =1

A set on which two operations are difined so as to have properties (A)-(E) called a field. The simplest possible field consist of two elements which we denote by 0 and 1, with addition defined by 0+0=1+(-1)=0, 1+0=0+1=1 (1) and multiplication defined by 0×0=0×1=1×0=0, 1×1=1 (2)

The order Relation

The real number system is ordered by relation <, which has the following properties:

6. For each pair of real numbers (a) and (b) exactly one of the following is true: a=b ,a<b or a>b

7. If a<b and b<c, then a<c (The relation < is transitive)

8. If a<b,thena+c<b+c for any c and if c>0, then a×c<b×c.

A field with an order relation satisfying (F)-(H) in an ordered field.

Theorem 1 (The Triangle Inequality). If aandb are any two real numbers then:

|a+b|≤|a|+|b| (3)

Corollary 2. If a and b are any two real numbers then |a-b|≥||a|-|b|| (4) and |a+b|≥||a|+|b|| (5) supremum of a set:

A set S of real numbers is bounded above if there is a real number b such that x≤b whenever x S. In this case, b is an upper bound of S. If b is an upper bound of S, then so is any large number, because of property (6)

If is an upper bound of S, but no number less than β is, then β is a supremum of S and we write β=supS

Example: If S is the set of negative numbers, then any nonnegative number is an upper bound of S,and sup=0

1.2 If S₁ is the set of negative integers,then any number a such that a≥-1 is an upper bound of S₁ , sup=-1

This example shows that a supremumof a set may or may not be in the set, since S₁ , contains its supremum, but S does not.

A nonempty set is a set that has at least one number. The empty set, denoted by , is the set that has no numbers.

The Completeness Axiom.

1. If a nonempty set of reak numbers is bounded above, then it has a supremum.

Property (I) is called conpleteness and we say that the real number system is a complete ordered field.

Thorem 3. If a nonempty set S of real numbers is bounded above, then sups is the unique real number β such that (a) x≤β for all x in S.

(b) if ε>0 there is an x₀ in S such that x₀>β-ε

Proof: we first show that β=supS has properties (a) and (b). Since β is an upper bound of it must satisfy (a). Since any real number a less than β can be written as β-ε with ε=β-a>0, (b) is just another way of saying that no number less than β is an upper bound of S, β=supS (satisfies (a) and (b))

Now we show that, there cannot be more than one real number with properties (a) and (b). Suppose thatβ₁<β₂ and β₂ has property (b) thus, if ε>0, there is an x₀ in S such that x₀> ₂-ε. Then by taking ε=β₂-β₁ , we see that there is an x₀ in S such that x₀>β₂-(β₂-β₁)=β₁,

So β₁ cannot have property (a). Therefore, there cannot be more than one real number that satisfies both (a) and (b).

Some Natation: “x is a member of S”=>x S “x is not a member of S” => x S.

Theorem 4. (The ArchemedeanProperty ). If p and ε are positive, then nε>p for some integer n.

Proof: The proof is by contradiction. If the statement is false, p is an upper bound of the set S={| x | x = nε, n is a integer}. Therefore, S has a supremum β, by property (I). Therefore, nε≤β (b)

1.3For all integer whenever n is , (b) implies that (n+1)ε≤β and therefore nε≤β-ε for all integers n. Hence, β-ε is an upper bound of S. Since, β-ε<β, this contradicts the definition of β.

Infimum of a set.

A set S of real numbers is bounded below if there is a real number a such that x≥a, whenever x S. In this case, a is a lower bounded of S.

but no number greater than α is, then α is an infimum of S, and we write: α=infS.

Theorem 8. If nonempty set S of real numbers is bounded below, then infS is the unique real number α such that:

1. x≥α for all x in S.

2. If ε>0, there is an x₀ in such that x₀< α+ε

Proof: A set S in bounded, of there are numbersa and b such that a≤x≤b for all x in S.

A bounded nonempty set has a unique supremum and a unique infimum, and: infS≤supS.

A nonempty set S of real numbers is unbounded above if it no upper bound, or unbounded below, if it has no lower bound - < x <+ (7)

We call points of infinity. If (S)is a nonempty set of real numbers, we write: supS= (8)

to indicate that S is unbounded above, and infS=- (9)

to indicate that S is unbounded below.

1. If a is any real number, then: a+ = +a= , a- =- +a= - , = =0

2. If a>0, then: a = a= , a(- )=(- )a=-

3. If a<0 , then: a = a= , a(- )=(- )a=

We also define: + = =(- )(- )= and - - = (- )=-

Finally, we define | |=|- |= .