INTRODUCTION
A satisfactory discussion of the main concepts of analysis (such as convergence, continuity, differentiation, and integration) must be based on an accurately defined number concept. We shall not, however, enter into any discussion of the axioms that govern the arithmetic of the integers, but assume familiarity with the rational numbers (i.e., the numbers of the form 
, where 
and 
are integers and 
).
The rational number system is inadequate for many purposes, both as a field and as an ordered set. (These terms will be defined in Secs. 1.6 and 1.12.)
For instance, there is no rational 
such that 
. (We shall prove this presently.) This leads to the introduction of so-called "irrational numbers" which are often written as infinite decimal expansions and are considered to be "approximated" by the corresponding finite decimals. Thus the sequence 
"tends to 
." But unless the irrational number 
has been clearly defined, the question must arise: Just what is it that this sequence "tends to"?
This sort of question can be answered as soon as the so-called "real number system" is constructed.
1.1 Example
We now show that the equation
(1)
is not satisfied by any rational 
. If there were such a 
, we could write 
where 
and 
are integers that are not both even. Let us assume this is done. Then (1) implies
(2)
This shows that 
is even. Hence 
is even (if 
were odd, 
would be odd), and so 
is divisible by 
. It follows that the right side of (2) is divisible by 
, so that 
is even, which implies that 
is even.
The assumption that (1) holds thus leads to the conclusion that both 
and 
are even, contrary to our choice of 
and 
. Hence (1) is impossible for rational 
.
We now examine this situation a little more closely. Let 
be the set of all positive rationals 
such that 
and let 
consist of all positive rationals 
such that 
. We shall show that 
contains no largest number and 
contains no smallest.
More explicitly, for every 
in 
we can find a rational 
in 
such that 
, and for every 
in 
we can find a rational 
in 
such that 
. To do this, we associate with each rational 
the number
(3)
Then
(4)
If 
is in 
then 
, (3) shows that 
, and (4) shows that 
. Thus 
is in 
.
If 
is in 
then 
, (3) shows that 
, and (4) shows that 
. Thus 
is in 
.
1.2 Remark
The purpose of the above discussion has been to show that the rational number system has certain gaps, in spite of the fact that between any two rationals there is another: If 
then 
. The real number system fills these gaps. This is the principal reason for the fundamental role which it plays in analysis.
In order to elucidate its structure, as well as that of the complex numbers, we start with a brief discussion of the general concepts of ordered set and field.
Here is some of the standard set-theoretic terminology that will be used throughout this book.
1.3 Definitions
If 
is any set (whose elements may be numbers or any other objects), we write 
to indicate that 
is a member (or an element) of 
. If 
is not a member of 
, we write: 
.
The set which contains no element will be called the empty set. If a set has at least one element, it is called nonempty.
If 
and 
are sets, and if every element of 
is an element of 
, we say that 
is a subset of 
, and write 
, or 
. If, in addition, there is an element of 
which is not in 
, then 
is said to be a proper subset of 
. Note that 
for every set 
.
If 
and 
, we write 
. Otherwise 
.
1.4 Definition
Throughout Chap. I, the set of all rational numbers will be denoted by 
.
