This new edition aims to guide undergraduate students through the first year of their mathematics course. It provides a rigorous introduction to Analysis, which takes into account the difficulties students often face when making the transition from A-level mathematics to this higher level. Plenty of examples are provided, some of which have full, detailed solutions, and others which encourage the student to discover and investigate the ideas themselves. Hints are provided, but the book aims to build confidence and understanding in all topics. This second edition has two new substantial chapters, covering integration and powere series, and is updated throughout, taking into account changes in notation.