Exploring the Infinite

An Introduction to Proof and Analysis

Jennifer Brooks

• Publisher: Chapman and Hall/CRC
• Year: 2016
• Print ISBN:
  9781498704496
• eBook ISBN:
  9781498704526
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Exploring the Infinite addresses the trend toward

a combined transition course and introduction to analysis course. It

guides the reader through the processes of abstraction and log-

ical argumentation, to make the transition from student of mathematics to

practitioner of mathematics.

This requires more than knowledge of the definitions of mathematical structures,

elementary logic, and standard proof techniques. The student focused on only these

will develop little more than the ability to identify a number of proof templates and

to apply them in predictable ways to standard problems.

This book aims to do something more; it aims to help readers learn to explore

mathematical situations, to make conjectures, and only then to apply methods

of proof. Practitioners of mathematics must do all of these things.

The chapters of this text are divided into two parts. Part I serves as an introduction

to proof and abstract mathematics and aims to prepare the reader for advanced

course work in all areas of mathematics. It thus includes all the standard material

from a transition to proof" course.

Part II constitutes an introduction to the basic concepts of analysis, including limits

of sequences of real numbers and of functions, infinite series, the structure of the

real line, and continuous functions.

Features

• Two part text for the combined transition and analysis course
• New approach focuses on exploration and creative thought
• Emphasizes the limit and sequences
• Introduces programming skills to explore concepts in analysis
• Emphasis in on developing mathematical thought
• Exploration problems expand more traditional exercise sets

Part II

Foundations of Analysis

Analysis is the branch of mathematics that grows out of calculus and the study of real-valued functions of a real variable. It seeks to make rigorous all the notions of calculus (limit, continuous functions, integrals) and to generalize them to other settings. Thus modern analysis considers functions from ℝⁿ to ℝ^m, complex-valued functions of one or more complex variable, generalizations of the notions of derivative and integral, and vector spaces of functions and the mappings between them.

The aims of this part of the book are to explore the most fundamental notions in analysis and to gain some familiarity with the kinds of arguments used in the field. We thus focus on two main topics: limits of sequences and continuous functions.

Chapter 8

Sequences of Real Numbers

In this chapter, we assume the existence of the ordered field ℝ of real numbers, although we do not yet discuss or use the completeness of the real numbers. In the next chapter, we will discuss one way to construct the real number system from the rational number system. The reader who wishes to wait to use the real number system until it has been rigorously defined can essentially replace every occurrence of the word “real” in this chapter with the word “rational.” We have already carefully considered the ordered field properties for ℚ, and all the properties of the real number system used in this chapter are also properties of the rational number system.

8.1 The Limit of a Sequence

You probably first studied sequences in second-semester calculus as a prelude to learning about series of numbers, power series, and Taylor’s Theorem. At that time, you probably did not work with a formal definition of limit but rather were content with an intuitive understanding. The intuitive definition of limit says that the sequence a = {a_n} has limit L if the terms of a are as close as we please to L for n sufficiently large. If a has a limit, we say it is convergent. Otherwise we s...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Table of Contents
• Preface
• I Fundamentals of Abstract Mathematics
• II Foundations of Analysis
• A A Very Short Course on Python
• Index