24 Chapter 1 • The Real Number System(b) Let n E NF be fixed. Vm E NF we let p(m) denote the proposition
p(m): nm E NF. Then:
(1) p(l) is the statement n · 1 E NF, which is obviously true.
(2) Suppose p(k) is true. Then nk E NF. Now,n(k+l) =nk+n,
and since NF is closed under addition, n(k + 1) E NF. Therefore, p(k + 1) is
true. We have thus proved that p(k) =::}-p(k + 1).
Therefore, by the principle of mathematical induction, Vm E NF, p(m) is
true. That is, Vm E NF, nm EN, which means NF is closed under multiplica-
tion.
( c) Exercise 1. •ORDINARY NATURAL NUMBERS
The "ordinary natural numbers" 1, 2, 3, 4,... exist in any ordered field in
the following sense: suppose a E F, an ordered field. We define1 = the multiplicative identity of F
2=1+1
3=2+1
4=3+1We shall use the symbols 1, 2, 3, ... to represent the elements of NF,
regardless of the ordered field F. Thus, every ordered field contains the ordinary
natural numbers, or at least a copy of them. Another way of saying this .is that
the natural numbers may be considered as "embedded" in any ordered field. For
this reason, we shall hereafter discontinue using the symbol NF, and
use instead the generic symbol N to represent the set of all natural
numbers, regardless of the ordered field F in which they occur.
Theorem 1.3.9 (Alternate Principle^6 of Mathematical Induction)
Suppose that '\:In E N, p(n) is a proposition about n such that
( 1) p( 1) is trne, and
(2) '\:/k E N, if p(m) is true for all natural numbers m < k in N then p(k)
is true.
Then '\:In E N , p(n) is true.- This is often called "strong mathematical induction," although it is equivalent to ordinary
mathematical induction rather than stronger than it.