3.2 Closed Sets and Cluster Points 149Theorem 3.2.13 (Balzano-Weierstrass Theorem for Sets) Every
bounded infinite set of real numbers has a cluster point.Proof. Let A be a bounded infinite set of real numbers. Since A is an infi-
nite set, we can find a sequence {an} of different elements of A. Then {an} is a
bounded sequence, so by the Balzano-Weierstrass Theorem for sequences, {an}
has a convergent subsequence, {ank}. Say ank --7 L. Then , every neighborhood
of L contains ank for infinitely many k. Since the terms of {an} are all different,
every neighborhood of L contains at least one point of A other than L. Thus,
L is a cluster point of A. •
THE CLOSURE OF A SET
A given set A of real numbers is not necessarily closed; it need not contain
all its cluster points. We want to be able to "close" it; that is, adjoin to it
just enough points to make the resulting set closed. This set will be called the
"closure" of A. Our task will be made easier if we start with a slightly different
definition.
Definition 3.2.14 If A ~ IR, the closure of A is the set A (or Acl) defined
as the intersection of the collection of all closed sets containing A.
Theorem 3.2.15 (Basic Properties of the Closure of A) \fA ~IR,
(a) A is a closed set;
(b) A~A;(c) A is the smallest closed set containing A, in the sense that if B is any
closed set containing A, th.en A~ B;(d) A is closed iff A= A;
(e) 0 = 0; IR =IR.
Proof. (a) By the closed set t heorem (3.2.4), the intersection of any col-
lection of closed sets is closed, so A is closed.
(b)- (e) Exercise 13. •
The interior of a set and the closure of a set share a duality relationship.
The interior of A is the largest open set contained in A, while the closure of
A is the smallest closed set containing A. For other interesting aspects of this
duality, compare for yourself the properties of interior and the properties of
closure listed in Theorems 3.1.11 and 3.2. 15.