# Exploring the Infinite

## An Introduction to Proof and Analysis

### About the author

**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.

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

300 pages; ISBN 9781498704502

Read online, or download in secure PDF format

Title: Exploring the Infinite

Author: Jennifer Brooks

- News
- Contents

### In the press

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*