In Section 0.3 we introduced the concept of a limit informally and relied on your intuition to understand what we said about limits in our development of Order Notation. In analyzing the qualitative behavior of sequences (Section 5.6) we made informal use of facts about the limits of products and quotients to determine longterm behavior. In Chapter 6, Section 6.4, we used the idea of a limit once again in Example 7 to show how to develop the formula for the Poisson distribution. In this appendix we wish to formalize the definition of a limit and some of the theorems related to it. The definitions and proofs in this appendix are covered in some calculus courses and all real analysis courses.

In discrete mathematics the main interest in limits is in connection with sequences. Only occasionally do we mention limits of functions (e.g., in Section 0.3), but since the two kinds of limits are closely related and because many readers of this book will be familiar with limits of functions from a calculus course, we ll define both kinds of limits here. We begin with limits of sequences.

Definition 1

Let {a _n } be a sequence of real numbers. We say that the sequence converges to a limit L, and write

if, for each positive number ?>0, there exists a positive integer N such that

In words, L is the limit of the sequence {a _n } if after some point (i.e., after some...