1549901369-Elements_of_Real_Analysis__Denlinger_

2.8 *countable and Uncountable Sets 127

Theorem 2.8.5 There is a sequence whose range is Q. That is, the set of

rational numbers can be arranged as a subsequence of a sequence.

Proof. Part 1: First, we show how to list all the positive rational numbers

in a sequence. List in a (horizontal) row all the positive rational numbers with

denominator 1, then in another row all those with denominator 2, then all

those with denominator 3, and so on. Then construct a sequence by following

the arrows in the pattern shown below.

Following the arrows will produce a sequence whose terms include all the

positive rational numbers: P1, p2, · · ·, Pn, Pn+1, · · ·.

etc.

Figure 2.8

Part 2: The following sequence will include all the rational numbers, pos-
itive, negative, and zero: 0, P1, -pi, P2, -p2, · · ·, Pn, -pn,Pn+1, -Pn+1, · · · ·

Corollary 2.8.6 The set Ql of rational numbers is countable.

Proof. In Theorem 2.8.5 we produced a sequence whose range is Ql; that
is, a function f from N onto Ql. This sequence includes many repetitions; i.e.,