viii Preface with "tips"on writing,ranging from the (what should be)obvious to the insider's preference ("Don't start a sentence with a symbol."). Proving.How can someone learn to prove mathematical results?There are many theories on this.We believe that learning mathematics is the same as learning to play an instrument or learning to succeed at a particular sport.Someone must provide the background:the tips,information on the basic skills,and the insider's"know-how." Then the student has to practice.Musicians and athletes practice hours a day,and it's not surprising that most mathematicians do,too.We will provide students with the background;the exercises and problems are there for practice.The instructor observes,guides,teaches and,if need be,corrects.As with anything else,the more a student practices,the better she or he will become at solving problems. Using this book.What should be in a book like this one?Even a quick glance at other texts on this subject will tell you that everyone agrees on certain topics:logic, quantifiers,basic set theoretic concepts,mathematical induction,and the definition and properties of functions.The depth of coverage is open to debate,of course.We try to cover logic and quantifiers fairly quickly,because we believe that students can only fully appreciate the fundamentals of mathematics when they are applied to interesting problems. What is also apparent is that after these essential concepts,everyone disagrees on what should be included.Even we prefer to vary our approach depending on our students.We have tried to provide enough material for a flexible approach. .The Minimal Approach.If you need only the basics,cover Chapters 1-18.(If you assume the well ordering principle,or decide to accept the principle of mathe- matical induction without proof,you can also omit Chapters 12 and 13.) .The Usual Approach.This approach includes Chapters 1-18 and Chapters 21- 24.(This is easily doable in a standard semester,if the class meets three hours per week.) .The Algebra Approach.For an algebraic slant to the course,cover Chapters 1-18, omitting Chapter 13 and including Chapters 27 and 28. The Analysis Approach.For a slant toward analysis,cover Chapters 1-23.(In- clude Chapter 24,if time allows.This is what we usually cover in our course.) Include as much material from Chapters 25 and 26 as time allows.Students usu- ally enjoy an introduction to metric spaces. Projects.We have included projects intended to let students demonstrate what they can do when they are on their own.We indicate prerequisites for each project,and have tried to vary them enough that they can be assigned throughout the semester.The results in these projects come from different areas that we find particularly interesting.Students can be guided to a project at their level.Since there are open-ended parts in each project,students can take these projects as far as they want.We usually encourage the students to work on these in groups. Notation.A word about some of our symbols is in order here.In an attempt to make this book user-friendly,we indicate the end of a proof with the well-known symbol The end of an example or exercise is designated by O.If a problem is used later in the text,we designate it by Problem".We also have a fair numberviii Preface with “tips” on writing, ranging from the (what should be) obvious to the insider’s preference (“Don’t start a sentence with a symbol.”). Proving. How can someone learn to prove mathematical results? There are many theories on this. We believe that learning mathematics is the same as learning to play an instrument or learning to succeed at a particular sport. Someone must provide the background: the tips, information on the basic skills, and the insider’s “know-how.” Then the student has to practice. Musicians and athletes practice hours a day, and it’s not surprising that most mathematicians do, too. We will provide students with the background; the exercises and problems are there for practice. The instructor observes, guides, teaches and, if need be, corrects. As with anything else, the more a student practices, the better she or he will become at solving problems. Using this book. What should be in a book like this one? Even a quick glance at other texts on this subject will tell you that everyone agrees on certain topics: logic, quantifiers, basic set theoretic concepts, mathematical induction, and the definition and properties of functions. The depth of coverage is open to debate, of course. We try to cover logic and quantifiers fairly quickly, because we believe that students can only fully appreciate the fundamentals of mathematics when they are applied to interesting problems. What is also apparent is that after these essential concepts, everyone disagrees on what should be included. Even we prefer to vary our approach depending on our students. We have tried to provide enough material for a flexible approach. • The Minimal Approach. If you need only the basics, cover Chapters 1–18. (If you assume the well ordering principle, or decide to accept the principle of mathe￾matical induction without proof, you can also omit Chapters 12 and 13.) • The Usual Approach. This approach includes Chapters 1–18 and Chapters 21— 24. (This is easily doable in a standard semester, if the class meets three hours per week.) • The Algebra Approach. For an algebraic slant to the course, cover Chapters 1–18, omitting Chapter 13 and including Chapters 27 and 28. • The Analysis Approach. For a slant toward analysis, cover Chapters 1–23. (In￾clude Chapter 24, if time allows. This is what we usually cover in our course.) Include as much material from Chapters 25 and 26 as time allows. Students usu￾ally enjoy an introduction to metric spaces. • Projects. We have included projects intended to let students demonstrate what they can do when they are on their own. We indicate prerequisites for each project, and have tried to vary them enough that they can be assigned throughout the semester. The results in these projects come from different areas that we find particularly interesting. Students can be guided to a project at their level. Since there are open-ended parts in each project, students can take these projects as far as they want. We usually encourage the students to work on these in groups. • Notation. A word about some of our symbols is in order here. In an attempt to make this book user-friendly, we indicate the end of a proof with the well-known symbol ut. The end of an example or exercise is designated by . If a problem is used later in the text, we designate it by Problem#. We also have a fair number