Exploring the Infinite

Exploring the Infinite addresses the trend toward a combined transition course and introduction to analysis course. Itguides the reader through the processes of abstraction and log-ical argumentation, to make the transition from student of mathematics topractitioner 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 limitsof sequences of real numbers and of functions, infinite series, the structure of the real line, and continuous functions. FeaturesTwo part text for the combined transition and analysis courseNew approach focuses on exploration and creative thoughtEmphasizes the limit and sequences Introduces programming skills to explore concepts in analysisEmphasis in on developing mathematical thoughtExploration problems expand more traditional exercise sets