Remember our discussion in Chapter 3 of how every real number has a dec-imal expression. We said that the expression b0.b1b2b3 . . . represents the realnumber that is the “sum to infinity” of the series b0 + b110 +

b2102 + b3103 + ∙ ∙ ∙

In this chapter we aim to make this statement precise. To do so, we need tointroduce one of the most fundamental concepts in mathematics – that of thelimit of a sequence of real numbers.First we need to say exactly what we mean by a sequence. That’s easyenough: a sequence is just an infinite list a1,a2,a3, . . . ,an, . . . of real numbersin a definite order. The number an is called the nth term of the sequence. Weusually denote such a sequence just by the symbol (an). Example 23.1 1. 1,1,1, . . . is the sequence (an) where an = 1 for all n.