Overview

In the early sections of this chapter we introduce the idea of the limit of a sequence and prove basic theorems on limits. The concept of a limit is central to subsequent chapters of this book. The later sections are devoted to the general principle of convergence and more advanced concepts of limits superior and limits inferior of a sequence.

10.1 The Concept of a Limit

If we observe the behaviour of several sequences, for example

(10.1)

(10.2)

(10.3)

(10.4)

we see that as n becomes larger the terms of each sequence approach a certain number: for Sequences (10.1), (10.2) and (10.4) it is 0, for (10.3) it is 1. In Sequences (10.1), (10.2), (10.3) and (10.4) there is a clear pattern in this approach. In the next example the terms of the sequence approach a certain number quite irregularly. The following table is a record of an experiment: casting of two dice and noting as a success whenever the sum is 7. The columns headed n in Table 10.1 indicate the number of throws, and the other columns gives the values of s/n where s is the number of successes in n trials.