Introduction to the Real Number System

Rosenlict's approach
The real numbers are basic to analysis, so we must have a clear idea of what they are. It is possible to construct the real number system in an entirely rigorous manner, starting from careful statements of a few of the basic principles of set theory, * but we do not follow this approach here for two reasons. One is that the detailed construction of the real numbers, while not very difficult, is time-consuming and fits more properly into a course on the foundations of arithmetic, and the other reason is that we already "know" the real numbers and would like to get down to business. On the other hand we have to be sure of what we are doing. Our procedure [...] is therefore to assume certain basic properties (or axioms) of the real number system, all of which are in complete agreement with our intuition and all of which can be proved easily in the course of any rigorous construction of the system. We then sketch how most of the familiar properties of the real numbers are consequences of the basic properties assumed and how these properties actually completely determine the real numbers.
See ita-2-0-1-pg15.

I will follow Rosenlicht's approach in ita chapter two, except in the following respects:

I assume that we do not already know the real numbers.
I assume that the properties of the real numbers are unfamiliar to us.

I make these assumptions because, I think ...
It is useful to assume that we know absolutely nothing about numbers and their properties.
We don't even know whether or not there are any numbers.
Then we can do an exercise in logical thought to gain a deeper understanding of numbers.

* The standard procedure for constructing the real numbers is as follows: One first uses basic set theory to define the natural numbers {1,2,3,...} (which, to begin with, are merely a set with an order relation), then one defines the addition and multiplication of natural numbers and shows that these operations satisfy the familiar rules of algebra. Using the natural numbers, one then defines the set of integers [ {...,-2,-1,0,1,2,...}] and extends the operations of addition and multiplication to all the integers, again verfying the rules of algebra. From the integers one next obtains the rational numbers or fractions. Finally, from the rational numbers one constructs the real numbers, the basic idea in this last step being that a real number is something that can be approximated arbitrarily closely by rational numbers. (The manufacture of the real numbers may be witnessed in E. Landau's Foundations of Analysis.) See ita-2-0-1-pg15.