This is the first of my book reviews, aimed at current undergraduate mathematicians, especially those at Warwick University.
For my first review, I will be looking at one of the recommended books (nb. link only works for Warwick Mathematicians) for Warwick's 1st year Analysis I & II course - Numbers and Functions: Steps to Analysis by R. P. Burn.
This book covers all areas of basic analysis and presents them in a problems-and-answers manner. That is to say, the majority of this book is written as brief introductory paragraphs followed by a number of questions which both test understanding of concepts, and produce methods of proving many of the Theorems the book gives. All problems have at least outline solutions, so if you get stuck there are always hints to push you in the right direction.
One of the major advantages of this book, at least for me in my first year, was that I felt that I was teaching myself these topics. As any first year student knows, lectures can be initially very daunting, and most people will find that if they are unable to understand one topic from the lecture, very quickly they will begin to slow down and lose understanding of the following concepts. The only way to remedy this is work outside of a lecture, and for me at least this did not happen. This book really saved me come revision time, and I used it to essentially learn the majority of the course at my own pace.
That is not to say that this book can replace an entire Analysis course - I would suggest you do rather more work than I did, especially if you have problems - but it is a very strong learning aid. It gives problems which are more engineered to gaining a greater understanding of a concept than a typical assignment sheet is, and will help you to not only remember the concepts, but to understand them more fully.
Come revision time, each chapter is finished off with a summary of the definitions and results from the chapter, and has question number references to help you pinpoint the area in which it was proven or used. The historical notes at each chapter are nice, but not really of much use (although they can be a welcome break after looking over 70-odd questions!).
On to content, the book is split into two main sections "Numbers" and, unsurprisingly, "Functions." It all starts very simply, harking back to A-level maths, with a more "mathsy" look at Induction and Inequalities. These give a nice slow introduction to some of the rigour of mathematics, and provide the reader with a real bridge to university-level maths. Sequences, Series and Completeness finish the Numbers section (totalling 131 pages and 352 questions!) and provide detailed looks at many concepts and theorems such as the Fundamental Theorem of Arithmetic, the Bolzano-Weierstrass Theorem and Power Series.
The second half of the book - "Functions" - concerns itself with continuity, completeness, calculus and function sequences. Weighing in at almost 200 pages and 417 questions, the majority of the book lies here. There are far too many concepts to really be mentioned here, but they are again presented in a brilliant style in which the reader really teaches themselves these concepts.
This book really, in my opinion, is the leading book for a first year analysis course, and outshines Guide to Analysis (Macmillan Mathematical Guides) by Mary Hart, and was truly a lifesaver for me saving me having to retake my Analysis exams.
This book is available from Amazon.co.uk from the link below - enjoy!