Advice to the reader

Mathematics cannot be learned well in a passive way. When you read this book, have paper and pen(cil) to hand: there are bound to be places where you cannot see all the details in your head, so be prepared to stop reading and start writing. Ideally, you should proceed as follows. When you come to the statement of a theorem, pause before reading the proof: do you find the statement of the result plausible? If not, why not? (try to disprove it). If so, then why is it true? How would you set about showing that it is true? Write down a sketch proof if you can: now try to turn that into a detailed proof. Then read the proof we give. Exercises The exercises at the end of each section are not arranged in order of difficulty, but loosely follow the order of presentation of the topics. It is essential that you should attempt a good portion of these. Understanding the proofs of the results in this book is very important but so also is doing the exercises. The second-best way to check that you understand a topic is to attempt the exercises. (The best way is to try to explain it to someone else.) It may be quite easy to convince yourself that you understand the material: but attempting the relevant exercises may well expose weak points in your comprehension.