Introduction to Mathematical Analysis I (Lafferriere, Lafferriere, and Nguyen)

Our goal with this textbook is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.

Tools for Analysis

This chapter discusses various mathematical concepts and constructions which are central to the study of the many fundamental results in analysis. Generalities are kept to a minimum in order to move quickly to the heart of analysis: the structure of the real number system and the notion of limit. The reader should consult the bibliographical references for more details.

The Natural Numbers and Mathematical Induction

We will assume familiarity with the set of natural numbers, with the usual arithmetic operations of addition and multiplication on , and with the notion of what it means for one natural number to be less than another. In addition, we will also assume the following property of the natural numbers.

Ordered Field Axioms

In this book, we will start from an axiomatic presentation of the real numbers. That is, we will assume that there exists a set, denoted by , satisfying the ordered field axioms, stated below, together with the completeness axiom, presented in the next section. In this way we identify the basic properties that characterize the real numbers. After listing the ordered field axioms we derive from them additional familiar properties of the real numbers. We conclude the section with the definition of absolute value of a real number and with several results about it that will be used often later in the text.

The Completeness Axiom for the Real Numbers

There are many examples of ordered fields. However, we are interested in the field of real numbers. There is an additional axiom that will distinguished this ordered field from all others. In order to introduce our last axiom for the real numbers, we first need some definitions.

Applications of the Completeness Axiom

We prove here several fundamental properties of the real numbers that are direct consequences of the Completeness Axiom.