Exploring the Infinite: An Introduction to Proof and Analysis

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