Preface xv Chapter 3. Individual instructors should feel free to pick a combination of chapters that best suits their students and their personal inclinations. I could also see an instructor using a single chapter as an enrichment for a standard course in the curriculum. For example, I could also see a course where Chapter 5 is covered while simultaneously teaching the students about metric spaces. Another way I have used this material is as a source for read- ing courses for undergraduates. Teaching this type of course is a challenge and a pleasure. The pleasure arises since you can let your tastes dictate the topics and the pace. This is a course where how much you cover is not so important as imparting a perspective and making a point. The challenge arises with the varied students who are likely to be seated in front of you. Unless you are at a small school where almost all the students follow the same path in mathematics, fashioning a course like the one this book was meant to support is going to be more work than teaching a course in one of the usual topics. I think it inevitable that when you use a book like this one and your goal is the same as the book’s and you have students with the same diverse backgrounds as I am used to seeing, then you will have to work a little harder than normal to keep the class together. Nevertheless, if you enjoy teaching, it will be rewarding. It is the teacher’s role to try to fill in gaps, prod students to push themselves, and adjust the pace of the material to suit the audience. The object in writing this book was not to present the material, but to teach the student. There are things I do here that I would never do in a monograph. For example, some things are repeated partially this is done to increase the independence of the chapters, but also repetition is frequently helpful and instructive. In addition, because this book is directed at undergraduates, I wanted to teach them how to learn mathematics. This is reflected in many ways. Some are subtle like the level of detail, talk about intuition, or what I think of as encouragement. Others are more blatant like the frequent insertion into the text of “(Why?)” and “(Verify.)” I also leave many details and routine arguments to the reader there is no better way to fix ideas in the brain than to carry these out. Advice to the Student Here are a few pointers for reading this book or any mathematics book. First, read with a writing implement and paper nearby. Just reading the words will not suffice you must fill in details and for that you must write. There is some reading of mathematics where you are only interested in getting the “big picture.” But while you are a student at the level at which this book is directed, you should be reading and understanding every detail. (And I do mean every.) You need to develop the skill of reading mathematics.
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