The Basics of Proofs
2.5 Basic Facts About Real Numbers .1 Ordered Fields
Many of the theorems of Calculus involve properties of the real numbers. Some of these properties are subtle, so it is essential to understand this important set of numbers. Already introduced are the sets of natural numbers,N, and the integers, Z.
Also of importance is the set of rational numbers,Q D fmnj m; n 2 Z; n ¤ 0g. This definition comes with the understanding that the two rational numbers mn andab are equal whenever mb D na. Thus, there are always infinitely many representations for each rational number. For all rational numbers r ¤0, one can always find relatively prime integers m and n with n> 0 such that r D mn. Together with an agreement to write the rational number 0 as 01, each rational number has a unique lowest terms representation.
The set of rational numbers is more than a set of fractions with integers for numerators and denominators. It also comes with the two binary operations of addition (C) and multiplication () and with the order relation less than (<).
The binary operations satisfy conditions which make Q into a field. A field F is a set with operations of addition and multiplication that satisfies the following axioms.
Axioms for a FieldF
A setF together with the binary operations of addition .C/ and multiplication ./ form a field if F contains the two elements 0 and 1 with 0 ¤ 1 such that for every r; s; t 2 F
rC s 2 F and r s 2 F and
rD s ! r C t D s C t rD s ! r t D s t the Closure Properties .r C s/ C t D r C .s C t/ .r s/ t D r .s t/ the Associative
Properties
rC s D s C r r s D s r the Commutative
Properties
rC 0 D r r 1 D r the Identity
Properties There exists r 2F If r ¤0, there exists 1r 2 F
such that r C.r/ D 0 such that r 1r D 1 the Inverse Properties
r .s C t/ D r s C r t the Distributive
Law of Multiplication Over Addition Notice that the rational numbers do satisfy the eleven field axioms. One defines the operation subtraction () by r s D r C.s/ and the operation division ( ) for s¤ 0 by r s D r 1s D rs. Moreover, the fieldQ together with the less than order relation is an ordered field that obeys the following axioms.
Axioms for an Ordered FieldF
A fieldF is an ordered field with order relation < if for every r; s; t 2 F exactly one of the following holds
r< s, r D s, s < r the Trichotomy Property r< s and s < t imply r < t the Transitive Property r< s implies r C t < s C t the Addition Property of
Less Than r< s and 0 < t imply r t < s t The Multiplication
Property of Less Than
2.5 Basic Facts About Real Numbers 33 Notice that the rational numbers do satisfy the four ordered field axioms. One defines the other order relations of greater than (>), greater than or equal to (), and less than or equal to () in the obvious ways, that is, r > s whenever s< r, r s whenever either r > s or r D s, and r s whenever either r < s or rD s.
There are many other ordered fields, and it is constructive to consider how to justify the fifteen ordered field axioms for a different ordered field. For example, the set T D fr C sp
2 j r; s 2 Qg is an ordered field using the usual addition and multiplication operations. For two elements a C bp
2 and c C dp relation you would want.a C bp
2/ < .c C dp
2/ whenever a c < .d b/p 2 which can be checked by squaring both a c and.d b/p
2, although you will need to also consider the signs of a c and d b. Thus, the definition becomes .a C bp
2/ < .c C dp
2/ if one of the following holds:
• a c< 0 and 0 < d b,
• 0 < a c, 0 < d b, and .a c/2< 2.d b/2, or
• a c< 0, d b < 0, and .a c/2> 2.d b/2.
It is fairly easy to check that T is an ordered field. The only field axiom which does not follow immediately from the properties of rational numbers is the inverse axiom for multiplication. You should verify that for a C bp
2 ¤ 0, its multiplicative
which is in T. The order axioms take more work to verify due to the complicated definition of less than. For example, to verify the less than relation works correctly with addition, one would begin with three elements of T, a C bp
2, c C dp
2/. To do this, one compares the values of.aCe/.cCe/ D ac and .d Cf /.bCf / D d b. But this reduces to comparing a c and d b which are known to satisfy the correct conditions because aC bp
2 < c C dp
2 was given.
Every ordered field satisfies a long list of simple properties that you will associate with facts learned in Arithmetic and Algebra. Here are some of those properties.
Some Properties Obeyed By All Ordered Fields Let r; s; t all be elements of ordered field F. Then 1. r 0 D 0.
2. If r C t D s C t, then r D s.
3. If r t D s t and t ¤0, then r D s.
4. .r/ D r.
5. If r ¤0, then 11 r
D r.
6. r D s if and only if r D s.
7. r D.1/ r.
8. .r/ C .s/ D .r C s/.
9. .r/ .s/ D r s.
10. If r< s and t < 0, then s t < r t.
11. If r ¤0, then r2 > 0.
12. 0 < 1.
13. If0 < r, then 0 < 1r.
14. If0 < r < s, then 0 < 1s < 1r.
15. If n is any natural number and r> 1, then rn< rnC1.
The reader may wish to prove some of these properties by applying the axioms.
This book will not dwell on these proofs since the techniques used in proving them are not essential for writing most proofs in Analysis. Two simple proofs are given here as examples.
PROOF: If r is any element of the fieldF, then r 0 D 0.
• Let r be an element of fieldF.
• Since 0 is the additive identity ofF, 0 D 0 C 0.
• Then r 0 D r .0 C 0/.
• By the Distributive Law, r 0 D r 0 C r 0.
• By adding r 0 to each side of this equality, one gets 0 D r 0 r 0 D .r 0 C r 0/ r 0 D r 0 C .r 0 r 0/ D r 0 C 0 D r 0.
• Therefore, for any r 2F, r 0 D 0.
The next theorem essentially says that if.1/ r has the same properties as r, it must equal r.
2.5 Basic Facts About Real Numbers 35
PROOF: If r is any element of the fieldF, then r D .1/ r.
Let r be an element of fieldF. Then
.1/ r D .1/ r C 0 Additive Identity
= .1/ r C .r C r/ Additive Inverses
= Œ.1/ r C r C r Associative Law of Addition
= Œ.1/ r C 1 r C r Multiplicative Identity
= Œ.1/ C 1 r C r Distributive Law
= 0 r C r Additive Inverses
= 0 C r r 0 D 0
= r Additive Identity
Therefore,.1/ r D r for every r 2 F.
Note that every ordered fieldF will contain a copy of Q. This follows since 0; 1 2 F, and if n is a natural number in F, then n C 1 2 F. Thus, it follows by mathematical induction that n 2F for all n 2 N. Moreover, since 0 < 1, it follows for each n 2 N that n D n C 0 < n C 1 showing that all natural numbers are distinct elements ofF. The existence of the negatives of all numbers in F implies that the integers is a subset ofF, and the existence of reciprocals implies that all of Q lies in F. There are fields which are not ordered fields, and some of them do not contain copies ofQ. Indeed, there are finite fields as well as infinite fields that do not containQ or even N.
2.5.2 The Completeness Axiom and the Real Numbers
There are infinitely many ordered fields. The real numbers,R, is special because it includes every number that is considered a possible “distance from zero,” either positive, negative, or zero. An easy way to ensure thatR contains every possible distance is to require it to satisfy the Completeness Axiom. This axiom considers nonempty subsets of an ordered field,F, (actually, any ordered set would do). A subset S F is said to be bounded above if there is an M 2 F such that all x 2 S satisfy x M. In this case, M is called an upper bound of S. Similarly, S F is bounded below by lower bound K 2F if all x 2 S satisfy x K. If S F is both bounded above and bounded below, then S is said to be bounded. If M is an upper bound for a set S, and it is less than or equal to every upper bound of S, then M is the least upper bound of S. Similarly, if K is a lower bound for a set S, and it is greater than or equal to every lower bound of S, then K is the greatest lower bound of S. For example, if S is the interval.1; 5 D fx j 1 x < 5g, then 10, 6, andp
30
are all upper bounds of S, but5 is the least upper bound of S. Also, 2, 0, and 12 are all lower bounds of S, but1 is the greatest lower bound of S. One often uses the notation l.u.b..S/ or sup.S/ to represent the least upper bound or supremum of S and g.l.b..S/ or inf.S/ to represent the greatest lower bound or infimum of S.
Axioms for the Real Numbers
The real numbers, R, is an ordered field that satisfies The Completeness Axiom:
Every nonempty set S R which is bounded above has a least upper bound inR.
Note, for example, that the set S D fx 2Q j x2< 7g is a nonempty subset of Q which is bounded above by 4, 3, and 2.7, but there is no element ofQ which is a least upper bound of S. The set of real numbers, though, does contain a least upper bound of S, namelyp
7. The Completeness Axiom is sometimes called the Least