ABSTRACT
Analysis is distinguished as a branch of mathematics by its frequent appeals to the notion of limit. Differentiation, integration, and infinite summation are all applications of the limit concept. So far we have used this concept only twice, in showing the existence of square roots of positive numbers and in proving that any two complete ordered fields are isomorphic. Both of these proofs appealed directly to the completeness of the real numbers. We shall now develop a theory of limits for both real and complex numbers which will serve as a prototype for a more general theory in Chapter 14.
