Under the inspiration of David Fowler, driven by the leadership of David Epstein, eased by the teaching ideas of Alyson Stibbard and challenged by the research of Lara Alcock, a remarkable transformation of the way in which students begin to study analysis took place at Warwick University. The division of teaching time between lectures and problem-solving by students in class changed to give student problem-solving pride of place. Many of the problems the students tackled in this approach to analysis were taken from the first five chapters of the first edition of this book. The experience has suggested a number of improvements to the original text. This new edition is intended to embody these improvements. The most obvious is the playing down of the Peano postulates for the natural numbers and algebraic axioms for the real numbers, which affects chapter 1 and the beginning of chapter 2. The second is in the display of ‘summaries’, which now appear when a major idea has been rounded off, not just at the end of the chapter. The third is an increase in the number of diagrams, and the fourth is the introduction of simple (and I hope evocative) names for small theorems, so that they may be cited more readily than when they only have a numerical reference. There are also numerous smaller points not just about individual questions. In particular, there is a suggestion of two column proofs in establishing the rudimentary properties of inequalities in chapter 2, and least upper bounds now figure more substantially in chapter 4.