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.
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