The Real Numbers
This text has two goals: (1) to develop the foundations that underlie calculus and
all of post calculus mathematics and (2) to develop students’ ability to understand
definitions, theorems, and proofs and to create proofs of their own – that is, to
develop students’ mathematical sophistication.
The typical freshman and sophomore calculus courses are designed to teach
the techniques needed to solve problems using calculus. They are not primarily
concerned with proving that these techniques work or teaching why they work.
The key theorems of calculus are not really proved, although sometimes proofs are
given which rely on other reasonable, but unproved, assumptions. Here we will
give rigorous proofs of the main theorems of calculus. To do this requires a solid
understanding of the real number system and its properties. This first chapter is
devoted to developing such an understanding.
Our study of the real number system will follow the historical development of
numbers: We first discuss the natural numbers or counting numbers (the positive
integers), then the integers, followed by the rational numbers. Finally, we discuss
the real number system and the property that sets it apart from the rational number
system – the completeness property. The completeness property is the missing
ingredient in most calculus courses. It is seldom discussed, but without it, one
cannot prove the main theorems of calculus.
The natural numbers can be defined as a set satisfying a very simple list of
axioms – Peano’s axioms. All of the properties of the natural numbers can be
proved using these axioms. Once this is done, the integers, the rational numbers,
and the real numbers can be constructed and their properties proved rigorously.
To actually carry this out would make for an interesting but rather tedious course.
Fortunately, that is not the purpose of this course. We will not give a rigorous
construction of the real number system beginning with Peano’s axioms, although
we will give a brief outline of how this is done. However, the main purpose of
this chapter is to state the properties that characterize the real number system
and develop some facility at using them in proofs. The rest of the course will be