Bridge to Abstract Mathematics: Mathematical Proof and Structures

314 PROPERTIES OF NUMBER SYSTEMS Chapter 9

DEFINITION 5

An ordered field F is said to be Archimedean ordered if and only if for all a E F

and b > 0, there exists a positive integer n such that nb > a.

In other words, if an ordered field is Archimedean ordered, then regard-

less of how small b is and how large a is, a sufficient number of repeated

additions b to itself will exceed a. See Article 10.3, Example 1 for an ex-

ample of a non-Archimedean ordered field.

THEOREM 1

A complete ordered field F is necessarily Archimedean ordered.

Proof Let a E F and b > 0 be given. Proceeding indirectly, suppose that

nb 5 a for all positive integers n, and consider the subset S = {nb 1 n E N}

of F. Clearly S is nonempty, since b E S, and S is bounded above, by a.

Hence, by the completeness property, S has a least upper bound in F,

call it u. Now u - b is less than u, so u - b is not an upper bound

of S. Hence there is a positive integer n' such that u - b < n'b. Thus

(n' + 1) b > u. Since n' + 1 E N, this contradicts the fact that u is an

upper bound of S. Hence our initial assumption nb I a for all n E N

must be incorrect; we are forced to accept the statement nb > a for some

n E N, the desired conclusion.

COROLLARY 1 \

The real number field is Archimedean ordered.

For those who are familiar with the notion of a convergent sequence of

real numbers, note the following fact.

COROLLARY 2

The sequence {llnl n = 1,2, 3,.. .) converges to zero in R.

Proof Let E > 0 be given. We must show that there exists n E N such that

l/n < e, that is, 1 < ne. Since R is Archimedean ordered, by Theorem

1, we may let a = 1 and b = E in Definition 5 to obtain the desired

conclusion.

Thus we have a rigorous derivation of the fact that the familiar and
important sequence (l/nj tends to zero in R. When the topic of infinite
sequences is covered in second or third semester calculus, this fact is gen-
erally taken for granted and then used as the basis for deriving convergence
properties of a large number of other sequences. Thus Corollary 2 fills a
significant gap in the calculus experience of most students.
The result of Corollary 2 can also be used to derive another familiar
fact about R, namely between any two reals, there is a rational number.
This property, which intuitively may seem inconsistent with our discovery,
in Article 8.3, that Q has "fewer" elements than R, is often described by the
i statement "Q is dense in R." Its proof follows easily from Corollary 2 if