#
Exploring the Infinite

An Introduction to Proof and Analysis

## Preview

## Book Description

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

## Table of Contents

**Fundamentals of Abstract Mathematics**

**Basic Notions**

A First Look at Some Familiar Number Systems

Integers and natural numbers

Rational numbers and real numbers

Inequalities

A First Look at Sets and Functions

Sets, elements, and subsets

Operations with sets

Special subsets of R: intervals

Functions

**Mathematical Induction **

First Examples

Defining sequences through a formula for the n-th term

Defining sequences recursively

First Programs

First Proofs: The Principle of Mathematical Induction

Strong Induction

The Well-Ordering Principle and Induction

**Basic Logic and Proof Techniques **

Logical Statements and Truth Table

Statements and their negations

Combining statements

Implications

Quantified Statements and Their Negations

Writing implications as quanti ed statements

Proof Techniques

Direct Proof

Proof by contradiction

Proof by contraposition

The art of the counterexample

**Sets, Relations, and Functions **

Sets

Relations

The definition

Order Relations

Equivalence Relations

Functions

Images and pre-images

Injections, surjections, and bijections

Compositions of functions

Inverse Functions

**Elementary Discrete Mathematics**

Basic Principles of Combinatorics

The Addition and Multiplication Principles

Permutations and combinations

Combinatorial identities

Linear Recurrence Relations

An example

General results

Analysis of Algorithms

Some simple algorithms

Omicron, Omega and Theta notation

Analysis of the binary search algorithm

**Number Systems and Algebraic Structures **

Representations of Natural Numbers

Developing an algorithm to convert a number from base

10 to base 2.

Proof of the existence and uniqueness of the base b representation of an element of N

Integers and Divisibility

Modular Arithmetic

Definition of congruence and basic properties

Congruence classes

Operations on congruence classes

The Rational Numbers

Algebraic Structures

Binary Operations

Groups

Rings and fields

**Cardinality **

The Definition

Finite Sets Revisited

Countably Infinite Sets

Uncountable Sets

**Foundations of Analysis **

**Sequences of Real Numbers**

The Limit of a Sequence

Numerical and graphical exploration

The precise de nition of a limit

Properties of Limits

Cauchy Sequences

Showing that a sequence is Cauchy

Showing that a sequence is divergent

Properties of Cauchy sequences

**A Closer Look at the Real Number System **

R as a Complete Ordered Field

Completeness

Why Q is not complete

Algorithms for approximating square root 2

Construction of R

An equivalence relation on Cauchy sequences of rational

numbers

Operations on R

Verifying the field axioms

Defining order

Sequences of real numbers and completeness

**Series, Part 1 **

Basic Notions

Exploring the sequence of partial sums graphically and

numerically

Basic properties of convergent series

Series that diverge slowly: The harmonic series

Infinite geometric series

Tests for Convergence of Series

Representations of real numbers

Base 10 representation

Base 10 representations of rational numbers

Representations in other bases

**The Structure of the Real Line **

Basic Notions from Topology

Open and closed sets

Accumulation points of sets

Compact sets

Subsequences and limit points

First definition of compactness

The Heine-Borel Property

A First Glimpse at the Notion of Measure

Measuring intervals

Measure zero

The Cantor set

**Continuous Functions**

Sequential Continuity

Exploring sequential continuity graphically and numerically

Proving that a function is continuous

Proving that a function is discontinuous

First results

Related Notions

The epsilon-delta□ condition

Uniform continuity

The limit of a function

Important Theorems

The Intermediate Value Theorem

Developing a root-finding algorithm from the proof of the

IVT

Continuous functions on compact intervals

**Differentiation**

Definition and First Examples

Properties of Differentiable Functions and Rules for Differentiation

Applications of the Derivative

**Series, Part 2 **

Absolutely and Conditionally Convergent Series

The rst example

Summation by Parts and the Alternating Series Test

Basic facts about conditionally convergent series

Rearrangements

Rearrangements and non-negative series

Using Python to explore the alternating harmonic series

A general theorem

**A Very Short Course on Python **

Getting Stated

Why Python?

Python versions 2 and 3

Installation and Requirements

Integrated Development Environments (IDEs)

Python Basics

Exploring in the Python Console

Your First Programs

Good Programming Practice

Lists and strings

if . . . else structures and comparison operators

Loop structures

Functions

Recursion

## Author(s)

### Biography

**Jennifer Halfpap** is an Associate Professor in the Department of Mathematical Sciences at the University of Montana, Missoula, USA. She is also the Associate Chair of the department, directing the Graduate Program.

## Reviews

This book consists of two distinct sections. The first resembles a traditional introduction to proof (including counterexamples) and standard mathematical topics (sets, functions, number theory, some abstract algebra, etc.). The work could serve as a textbook for a semester course on that alone. The second part focuses on analysis of the real line. The work begins by establishing the existence of an uncountable set followed by the completion of the real line via Cauchy sequences. Next is the topology of the real line (basic point set in a metric space ending with Heine-Borel and the Cantor set). It concludes by examining continuous and uniformly continuous functions, derivatives, and absolutely and conditionally convergent series and rearrangements. The book is well written and accessible to students, with thought-provoking exercises sprinkled throughout and larger exercise sets at the end of each chapter. It could easily be used for a two-semester course after multivariable calculus, preparing students with the fundamentals for upper-division courses, particularly an advanced calculus course. In the appendix, there are also â€œProgramming Projects,â€ such as a brief course on Python as a suggested language. This book is worthy of consideration.

--J. R. Burke, Gonzaga University, Choice magazine 2016