A First Course in Optimization Theory

There are many methods for obtaining the real number system from the rational number system. We describe one in this appendix, which “constructs” the real number system as (appropriately defined) limits of Cauchy sequences of rational numbers. An alternative constructive approach—the method of Dedekind cuts—is described in Rudin (1976). A third approach, which is axiomatic, rather than constructive, may be found in Apostol (1967), Bartle (1964), or Royden (1968).

Our presentation in this appendix, which is based on Strichartz (1982), is brief and relatively informal. For omitted proofs and greater detail than we provide here, we refer the reader to Hewitt and Stromberg (1965), or Strichartz (1982).

Construction of the Real Line

We use the following notation: ℥ will denote the set of natural numbers and ℤ the set of all integers:

ℚ will denote the set of rational numbers:

It is assumed throughout this appendix that the reader is familiar with handling rational numbers, and with the rules for addition (+) and multiplication (·) of such numbers. It can be shown that under these operations, the rational numbers form a field; that is, for all rationals a, b, and c in ℚ, the following conditions are met:

1. Addition is commutative: a + b = b + a.

2. Addition is associative: (a + b) + c = a + (b + c).

3. Multiplication is commutative: a · b – b · a.

4. Multiplication is associative: (a · b) · c = a · (b · c).

5. […]