This textbook is part of a major series designed to guide undergraduate students through the first year of their mathematics course. Each book is designed to enable students to acquire confidence, to test their understanding and to achieve mastery of the subject they are studying. New concepts are introduced gradually with worked examples as illustrations. This text aims to minimize the difficulties which arise from the contrast between analysis and sixth form mathematics. It includes historical notes and anecdotes which will help the reader to appreciate how the subject developed to its present form. Exercises with and without solutions are provided.