Introductory real analysis can be an exciting course; it is the gateway to an impressive panorama of higher mathematics. But for all too many students, the excitement takes the form of anxiety or even terror; they are overwhelmed. For many, their study of mathematics ends one course sooner than they expected, and for many others, the doorways that should have been opened now seem rigidly barred. It shouldn’t have to be that way, and this book is offered as a remedy.
GOALS FOR INTRODUCTORY ANALYSIS
The goals of first courses in real analysis are often too ambitious. Students are expected to solidify their understanding of calculus, adopt an abstract point of view that generalizes most of the concepts, recognize how explicit examples fit into the general theory and determine whether they satisfy appropriate hypotheses, and not only learn definitions, theorems, and proofs but also learn how to construct valid proofs and relevant examples to demonstrate the need for the hypotheses. Abstract properties such as countability, compactness and connectedness must be mastered.