50 Chapter 2 • Sequences2.1 Basic Concepts: Convergence and Limits
A commonly-held, intuitive understanding of an (infinite) sequence is that it is
an "infinite succession of numbers,'' not necessarily different:Of course, this is merely a suggestive description. It cannot serve as a rigorous
definition, since it does not define what is meant by an "infinite succession" of
numbers. Another intuitive way of describing an infinite sequence is as a vector
with infinitely many components:Again, this description may help us feel more comfortable with the notion of
a sequence, but it fails to be a rigorous definition since it leaves "vector" and
"component" undefined.
Finally, we give a precise and rigorous definition.Definition 2.1.1 A sequence of real numbers is a function x: N -t R
That is, given any natural number n , there is a corresponding real number
x(n).Comments and some conventions:(1) We shall call x(n) the nth term of t he sequence.(2) We shall hereafter always write the nth term as Xn, using subscript no-
tation rather t han the functional notation x(n).(3) The sequence itself will be denoted { xn}, or occasionally { xn}~=l ·(4) Since by Definition 2.1.1, all sequences contain infinitely many1 terms, it
will not be necessary to call them infinite sequences. We merely call them
sequences.(5) Conventions (2) and (3) together make rigorous the intuitive view of a
sequence { Xn} as an infinite succession of numbers,- Strictly speaking, a sequence need not h ave infinitely many different terms, since some (or
a ll ) of them may be the same.