Exploring the Infinite

This is a textbook for an introductory course in analysis, combining topics from a transition course. At some schools, a transition course is combined over one or two semesters to introduce topics from real analysis. This allows students a more gradual approach to the difficult topics of analysis. Beginning with logic and sets, this text gradual

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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