Our goal in this chapter is to provide a quick review of a handful of important ideas from advanced calculus (and to encourage a bit of practice on these fundamentals). We will make no attempt to be thorough. Our purpose is to set the stage for later generalizations and to collect together in one place some of the notation that should already be more or less familiar. There are sure to be missing details, unexplained terminology, and incomplete proofs. On the other hand, since much of this material will reappear in later chapters in a more general setting, you will get to see some of the details more than once. In fact, you may find it entertaining to refer to this chapter each time an old name is spoken in a new voice. If nothing else, there are plenty of keywords here to assist you in looking up any facts that you have forgotten.

The Real Numbers

First, let's agree to use a standard notation for the various familiar sets of numbers. ℝ denotes the set of all real numbers; ℂ denotes the set of all complex numbers (although our major concern here is ℝ, we will use complex numbers from time to time); ℤ stands for the integers (negative, zero, and positive); ℕ is the set of natural numbers (positive integers); and ℚ is the set of rational numbers.