Unique in approach, this text is specifically designed to help today's weaker students achieve competence in serious mathematics. This text develops mathematical sophistication with a thorough knowledge and understanding of limiting processes. Unlike other texts — which merely present the theory and follow with a set of exercises in a routine fashion — this text provides insight into the techniques of analysis — explaining at the end of each key definition, example, and theorem what students must do to come to grips with the concepts. Concepts are presented slowly and include the details of calculations as well as substantial explanations as to how and why one proceeds in the given manner. Comprehensive in coverage, the text explores the principles of logic, the axioms for the real numbers, limits of sequences, limits of functions, differentiation and integration, infinite series, convergence, uniform convergence for sequences of real-valued functions — and treats several topics often slighted or omitted in other texts.