This lively introductory text introduces students to the benefits of a rigorous examination of functions of a real variable. Each chapter features informal discussions on questions that lend inner fascination to analysis, alongside precise but not overly formal developments of the techniques necessary for understanding them. By focusing on the connecting themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text transforms what could be an overwhelming cascade of definitions and theorems into a coherent and engaging flow of ideas. With the awareness that rigor is essential, students are much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of teaching experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition.