3. Inequalities and ordered ﬁelds 9

upper bound means the smallest possible upper bound; the concept juxtaposes

small and big. The least upper bound α of S is the smallest number that is greater

than or equal to any member of S. See Figure 1.2. One cannot prove that such

a number exists based on the ordered ﬁeld axioms; for example, if we work within

the realm of rational numbers, the set of x such that

x2

2 is bounded above, but

it has no least upper bound. Mathematicians often use the word supremum instead

of least upper bound; thus sup(S) denotes the least upper bound of S. Postulating

the existence of least upper bounds as in the next deﬁnition uniquely determines

the real numbers.

S

m1

α

m2

Figure 1.2. Upper bounds.

Deﬁnition 3.2. An ordered ﬁeld F is complete if whenever S is a nonempty subset

of F and S is bounded above, then S has a least upper bound in F.

We could have instead decreed that each nonempty subset of F that is bounded

below has a greatest lower bound (or inﬁmum). The two statements are equivalent

after replacing S with the set -S of additive inverses of elements of S.

In a certain precise sense, called isomorphism, there is a unique complete or-

dered ﬁeld. We will assume uniqueness and get the ball rolling by making the

fundamental deﬁnition:

Deﬁnition 3.3. The real number system R is the unique complete ordered ﬁeld.

3.2. What is a natural number? We pause to briefly consider how the natural

numbers ﬁt within the real numbers. In our approach, the real number system is

taken as the starting point for discussion. From an intuitive point of view we can

think of the natural numbers as the set {1, 1+1, 1+1+1,...}. To be more precise,

we proceed in the following manner.

Deﬁnition 3.4. A subset S of R is called inductive if whenever x ∈ S, then

x + 1 ∈ S.

Deﬁnition 3.5. The set of natural numbers N is the intersection of all inductive

subsets of R that contain 1.

Thus N is a subset of R, and 1 ∈ N. Furthermore, if n ∈ N, then n is an

element of every inductive subset of R. Hence n + 1 is also an element of every

inductive subset of R, and therefore n+1 is also in N. Thus N is itself an inductive

set; we could equally well have deﬁned N to be the smallest inductive subset of R

containing 1. As a consequence we obtain the principle of mathematical induction:

Proposition 3.1 (Mathematical induction). Let S be an inductive subset of N

such that 1 ∈ S. Then S = N.