10.3 OUTLINE OF THE CONSTRUCTION OF THE REALS 353to present are, in all likelihood, slightly beyond the level of experience
of most readers. Specifically, a full understanding of the construction of
R from Q, by using equivalence classes of Cauchy sequences, requires
experience and well-developed facility in working with properties of
sequences, and especially a working knowledge of the definitions of con-
vergent sequence and Cauchy sequence. In truth, most students who are
ever going to go through all the details of this construction should probably
do so after their course in advanced calculus.
For these reasons, we place even less emphasis in this article than before
on providing full proofs of most theorems. For the more readily acces-
sible arguments, we strike a balance between proofs given in the text and
those left as exercises. In the case of a number of either very detailed
and lengthy or quite technical proofs, we give only an outline or some dis-
cussion of the proof. If highly motivated, you may take this as a challenge
to your proof-writing ability, whereas if you are more casual in your ap-
proach, then you should only try to follow the main lines of the development.
So let us begin our sketch of the construction of R; we start with some
heuristics. Think of a familiar irrational number such as z. A familiar
characteristic of this number is its nonterrninating, nonrepeating decimal
expansion that begins 3.141 59. One way to think of the number n, using
rational numbers only, is as the limit of the sequence x, = 3.1, x, = 3.14,
x, = 3.141, and so on. This example illustrates the idea of an irrational
number as the limit of a sequence of rational numbers, and so introduces
us to the basic idea behind the construction of the reals from the rationals
by way of sequences. But, of course, the rules of the game do not permit
us to work back from a known irrational number to a sequence of rationals.
We must proceed outward from the rational number system, with no a priori
knowledge of any other type of number. The idea that should motivate
us, however, is this. It seems from the preceding example that there are
sequences of rationals that "want to" converge but have nothing to converge
to in the system Q. Theorem 1, one of the few we prove in this article,
gives us an initial handle on precisely which sequences of rationals "seem
to want" to converge. First, let us formulate a definition of a convergent
sequence of rational numbers.
DEFINITION 1
A sequence iqnj of rational numbers is
ber q, denoted q, -+ 9, if and only if, to
corresponds a positive integer N such
n 2 N.THEOREM 1said to converge to the rational num-
every positive rational number E, there
that (q, - ql < E whenever n E N withIf (q,) is a convergent sequence of rational numbers, then {q,,) satisfies the
condition: To every positive rational number 8, there corresponds a positive in-
teger N such that 19, - q,l < E whenever m, n E N with m, n > N.