When you first arrive at university to do a mathematics degree, ‘analysis’ seems like just another word. Unlike ‘calculus’ or ‘algebra’, it doesn’t have a particularly mathematical sound to it. ‘Analysis?’, I here you say when you’re first looking over your modules, ‘doesn’t sound too hard’. But of course after about two weeks you realise that it’s about the hardest thing you’ve ever done in your life. Well, at least mathematically.

To learn analysis you need talent, motivation, and good teaching. But having a first-rate text to refer to when you just can’t remember that theorem or need help with a tricky definition is very useful. Introduction to Real Analysis is a great book. It covers real analysis from its roots in the real numbers and sequences to its core topics of differentiation and integration. It even gives you a taste of topology at the end. Its exposition is clear, which makes it good for beginners, and thorough, which makes it good for more advanced students.

The trouble with some textbooks is that they rely too much on the reader completing exercises, which wouldn’t be a problem if we didn’t already have weekly workbooks or assignment sheets. Introduction to Real Analysis includes many exercises (and selected solutions) but it doesn’t rely on them. Many fully-solved examples are provided and theorems are punctuated with very clear – but not overly simplified – explanation of the material being covered.

Another good thing about the book is that all notation is explained. That might sound a little silly, but for a beginner the strange world of mathematical notation can be a daunting one, so having a book that tells you the meaning of each symbol in a straight-forward manner is very useful. It also has an appendix on logic and proofs, which is a useful reference when you’re first trying to get your head round ideas such as ‘if and only if’ and logical quantification.

At the beginning of each chapter is a short biographical account of the mathematician whose work is most relevant to the proceeding material, as well as some general explanation of the historical context in which the mathematics in question was developed. This might not be directly useful to learning analysis, but it is interesting and perhaps useful for the third-year course MA3E5 History of Mathematics.

All in all, Bartle and Sherbert have done an excellent job with this book. It is rigorous and yet understandable. I only wish I had had it in Term 1 of the first year; I’ve forgotten how many times I got totally stuck on an Analysis II assignment sheet question and reached for my trusty ‘green book of analysis’ for help. (The cover’s green, you see, although you probably already realised that, what the picture with and all…)