Introduction to Mathematical Analysis I - Second Edition

58 2.6 OPEN SETS, CLOSED SETS, COMPACT SETS, AND LIMIT POINTS has a convergent subsequence, { a n k } . Say, lim k → ∞ a n k = a . It now follows from Theorem 2.6.3 that a ∈ A . This shows that A is compact as desired. Definition 2.6.4 (cluster/limit/accumulation point). Let A be a subset of R . A point a ∈ R (not necessarily in A ) is called a limit point of A if for any δ > 0, the open ball B ( a ; δ ) contains an infinite number of points of A . A point a ∈ A which is not an accumulation point of A is called an isolated point of A. Example 2.6.5 (1) Let A = [ 0 , 1 ) . Then a = 0 is a limit point of A and b = 1 is also a limit point of A . In fact, any point of the interval [ 0 , 1 ] is a limit point of A . The set [ 0 , 1 ) has no isolated points. (2) Let A = Z . Then A does not have any limit points. Every element of Z is an isolated point of Z . (3) Let A = { 1 / n : n ∈ N } . Then a = 0 is the only limit point of A . All elements of A are isolated points. Example 2.6.6 If G is an open subset of R then every point of G is a limit point of G . In fact, more is true. If G is open and a ∈ G , then a is a limit point of G \ { a } . Indeed, let δ > 0 be such that B ( a ; δ ) ⊂ G . Then ( G \ { a } ) ∩ B ( a ; δ ) = ( a − δ , a ) ∪ ( a , a + δ ) and, thus B ( a ; δ ) contains an infinite number of points of G \ { a } . The following theorem is a variation of the Bolzano-Weierstrass theorem. Theorem 2.6.6 Any infinite bounded subset of R has at least one limit point. Proof: Let A be an infinite subset of R and let { a n } be a sequence of A such that a m 6 = a n for m 6 = n (see Theorem 1.2.7 ) . Since { a n } is bounded, by the Bolzano-Weierstrass theorem (Theorem 2.4.1 ) , it has a convergent subsequence { a n k } . Set b = lim k → ∞ a n k . Given δ > 0, there exists K ∈ N such that a n k ∈ B ( b ; δ ) for k ≥ K . Since the set { a n k : k ≥ K } is infinite, it follows that b is a limit point of A . The following definitions and results provide the framework for discussing convergence within subsets of R . Definition 2.6.5 Let D be a subset of R . We say that a subset V of D is open in D if for every a ∈ V , there exists δ > 0 such that B ( a ; δ ) ∩ D ⊂ V . Theorem 2.6.7 Let D be a subset of R . A subset V of D is open in D if and only if there exists an open subset G of R such that V = D ∩ G . Proof: Suppose V is open in D . By definition, for every a ∈ V , there exists δ a > 0 such that B ( a ; δ a ) ∩ D ⊂ V . Define G = ∪ a ∈ V B ( a ; δ a )