1549901369-Elements_of_Real_Analysis__Denlinger_

2.1 Basic Concepts: Convergence and Limits 51

or as a vector with infinitely many components,

{xn} = (x1, Xz, X3,... 'Xn, Xn+l, ... ).

(6) We must be careful not to let the braces in the notation mislead us into

thinking that we are talking about a set of numbers. For example, the

sequence { 1} consisting of infinitely many terms, each of which equals 1,

is different from the set { 1}, which contains only one element, 1. We use

the same notation {l} in both cases; the context will determine which

interpretation we mean.

Example 2.1.2 The first six terms of the sequence { 2 + (-~)n} are

X1=l, X2=2~,

X4 =^2 4,^1 X5 --^14 S'

X3 -12 - 3>

X5 --^21 6·

CONVERGENCE OF A SEQUENCE
The most important concept associated with sequences is that of conver-
gence to a limit. Intuitively, when we say that a sequence { Xn} converges to
limit L , we mean that as n gets larger and larger, without bound, the terms Xn
of the sequence get "close to" the number L; equivalently, the distance between
Xn and L, which we measure by lxn - LI, gets smaller than any positive real
number. If we plot the function x : N ~ JR in a coordinate system with a hori-
zontal n-axis and vertical y-axis (where y = Xn), then the statement that { Xn}
converges to limit L is equivalent to saying that this graph has the horizontal
line y =Las an asymptote. See Figure 2.l(a).

Example 2.1.3 We graph the sequence {xn} = { 2 + (-~)n} by plotting

the function y = Xn in a two-dimensional coordinate system. Observe in Fig-
ure 2.l(a) that the horizontal line y = 2 is an asymptote. We thus say that the
sequence {xn} converges to the number 2 as its limit.

x,,

.

2 --------;--~--.--J __ T __ ._ _____ _. __

2 3

I

/l

2 3 4 5 6 7 8 9 10

(a) (b)

Figure 2.1