1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 115

7. Let {Ga: a EI} be an open covering of the compact set ACC. Prove

that there exists an c > 0 such that for every zo E A, the neighborhood

Ne ( zo) is contained in at least one of the sets G °'. This property is called

Lebesgue's covering lemma, and c; is called a Lebesgue number for the

op en covering { G °'}.

8. Let A be a region in JR^2 and Ba compact subset of A. Show that there
   is always a simple closed polygon P contained in A and such that B is
   contained in the region bounded by P.

2.12 COMPLETENESS

Another important metric space concept is that of completeness, which we
proceed to discuss briefly in this section. As we shall see later, many results
in complex analysis depend on the completeness of C.

Definition 2.41 Let ( S, d) be a metric space and let { x n} be a sequence
in S. We say that {xn} is a fundamental sequence, of a. Cauchy sequence, iff
for each c, > 0 there is a positive number Ne such that m, n > Ne implies
that d( Xn, Xm) < c. - -

Theorem 2.25 Every convergent sequence in ( S,-d) is a Cauchy se_quence.
However, in some metric spaces not every Cauchy -sequence converges to
a point in the space.

Proof Suppose that {xn} ___, x. Then given c: > 0 there exists Ne such

that n > Ne implies that

d( Xn, X) <^1 / 2 c

Thus, if m, n > Ne,. we have

d(xn,xm)::; d(xn,x) + d(x,xm) <^1 / 2 c +^1 / 2 c = c:

which shows that {xn} is a Cauchy sequence.
To see that the converse may fail to hold in some spaces, let S
{ x : x E JR, 0 ::; x < 1} with the usual metric, and consider the sequence
Xn = (n-1)/n (n = 1,2, ... ). This is a Cauchy sequence since

. .. --;- 1 1 1 1
d(xn, Xm) = lxn - Xml =I-rn - -n I< -m + -n < c

by choosing m,n > Ne = [2/c:] + 1. Yet {xn} does not converge to a
point of S.

Definition 2.42 A metric space (S, d) is said to be complete iff every
Cauchy sequence in S converges to a point of S.
The concept of a complete metric space is due to Frechet and Hausdorff.