22 Chapter 1 • The Real Number System
Therefore, A is an inductive set. By Theorem 1.3.4, Np ~ A. That is, all
natural numbers in F are 2: 1.
( c) Let A = { n E Np : n > 1 =? n - 1 E Np}. Then
(i) 1 E A, since 1 j. 1.
(ii) Suppose x E A. Then x > 1 =? x - 1 E Np. Consider x + 1. Since
x E Np, x 2: 1, so, x + 1 > 1 and (x + 1) - 1 = x E Np. Thus,
x + 1 > 1 =? (x + 1) - 1 E Np.
Thus, x + 1 E A.
Therefore, A is an inductive set. By Theorem 1.3.4, Np C A. That is,
Vn E Np, if n > 1, then n - 1 E Np. •
A surprising result of this axiomatic approach is that it enables us to prove
the principle of mathematical induction. This important technique of proof is
itself a theorem within the theory of ordered fields. Most students regard the
principle of mathematical induction as an axiom of logic that must be assumed;
it may come as a surprise to you that it is actually a theorem.
Theorem 1.3.6 (The Principle of Mathematical Induction) Let F be
an ordered field. Suppose that Vn E Np, p(n) is a proposition about n. If
(1) p(l) is true, and
(2) Vk E Np, p(k) =? p(k + 1),
then Vn E Np, p(n) is true.
Proof. Suppose p(n) is as described in the hypotheses. Let A= {x E Np:
p(x) is true}. Then
(i) 1 EA, by (1).
(ii) Suppose x E A. Then x E Np and p(x) is true. Thus, by (2), p(x + 1)
is true. That is, x + 1 E A. Therefore, x E A =? x + 1 E A.
Therefore, A is an inductive set. By Theorem 1.3.4, Np ~ A. That is, Vn E
Np, p(n) is true. •
In the remainder of this section, we put the principle of mathematical induc-
tion to good use, as we derive some additional results about natural numbers.
Theorem 1.3. 7 Let F be an ordered field.
(a) Vm,n E Np, ifm < n, then n - m E Np.
(b) Vn E Np, there is no natural number between n and n + l.*Proof. (a) Vn E Np we let p(n) denote the proposition
p(n): Vm E Np, m < n =? n - m E Np.•An asterisk before a theorem, proof, or other item in this chapter indicates that the item is
challenging and can be omitted, especially in a one-semester course.