Nonlinear Control Sy.. - Free

2.6. SEQUENCES 45
Compact set: A set A c R" is said to be compact if it is closed and bounded.
Convex set: A set A C iR' is said to be convex if, whenever x, y E A, then
also belongs to A.
2.6 Sequences
9x1+(1-0)x2i 0 N implies that d(xn, x0) < . We then write
x0 = limxn, or xn -* x0 and call x the limit of the sequence {xn}.
It is important to notice that in Definition 2.12 convergence must be taking place in
the metric space (X, d). In other words, if a sequence has a limit, lim xn = x' such that
x` V X, then {xn} is not convergent (see Example 2.10).
Example 2.8 Let X1 = IR, d(x, y) = Ix - yl, and consider the sequence
{xn} = {1,1.4,1.41,1.414, }
(each term of the sequence is found by adding the corresponding digit in v'-2). We have that
f- x(n) l < e for n> N
and since i E IR we conclude that xn is convergent in (X1, d).
Example 2.9 Let X2 = Q, the set of rational numbers (x E Q = x = b, with a, b E 7G, b #
0). Let d(x, y) = Ix - yl, and consider the sequence of the previous example. Once again
we have that x, is trying to converge to f. However, in this case f Q, and thus we
conclude that xn is not convergent in (X2, d).
Definition 2.13 A sequence {xn} in a metric space (X, d) is said to be a Cauchy sequence
if for every real > 0 there is an integer N such that d(xn, x,n) < 1;, provided that n, m > N.
It is easy to show that every convergent sequence is a Cauchy sequence. The converse is,
however, not true; that is a Cauchy sequence in not necessarily convergent, as shown in the
following example.