This textbook 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 & others which encourage the student to discover & investigate the ideas themselves. Hints are provided but the book aims to build confidence & understanding in all topics. This new edition has two new substantial chapters covering integration & power series & is updated throughout taking into account changes in notation.