Answers & Hints for Selected Exercises 651( {:::) Suppose every real number is a cluster point of A. Let a < b. Let
c = a!b and c = b;a. Since c is a cluster point of A, NE:(c) must contain a
point x of A other than C. Then a< x < b. [Draw figure.]- (::::})Suppose A is dense in B. Let b EB and c > 0. Then b EA, so by Ex.
25, 3 sequence {an} in A converging to b. Then every NE: ( x) contains at least
one point of A.
( {:::) Suppose t hat Vb E B, every nbd of b contains a point of A. Let b E B.
Then Vn EN, 3 an EA 3 an E Nl.(b), sob EA. Thus B ~A; i.e., A is dense
in B. n
EXERCISE SET 3.3- Let U be an open cover of a finite set A = { a 1 , a 2 , .. · , an}. Then Vi =
1, 2, · · · , n , 3 Ui EU containing ai. Then {U 1 , U 2 , · · · , Un} is a finite subcover. - Part (a):
(a) {(-n,n): n EN}
(c) {(a+ ~,b+ ~): n EN}
(f) { (-n: a+~) : n EN}
(b) { (a + ~, b - ~) : n > b~a }
(e) { (-oo: a - ~) : n EN}
(i) {Nn~i (~) :nEN}
Part (b): Apply Thm. 3.3.6 or 3.3.8.
(a),(e),(f),(g),(h),(j),(k),(1) not bounded. (b),(c),(d),(e),(g),(i),(1) not closed.- If U is an open cover of the union of compact sets A 1 , A2, · · · , An, then each
set Ai can be covered by a finite number mi of sets in U. Then LJ~ 1 Ai can
be covered by m 1 + m 2 + · · · + mn (a finite number) of sets in U. - Suppose Xn --> L, and let S = {xn : n E N} U {L}. Since a convergent
sequence is bounded, S is bounded, and contains its only cluster point, L. So,
S is closed, and hence compact. - A is nonempty, compact ::::} A is bounded, nonempty ::::} 3 u = inf A and
v = sup A. By Ex. 3.2. 15 , u, v E A since A is closed. Then u = min A and
v = maxA (Thm. 1.6.5). - Let A be compact and B ={Ix -yl: x, y EA}.
First show B compact using sequential criterion. Let {bn} be a sequence in
B. Then Vn EN, 3 Xn, Yn EA 3 bn = lxn - Ynl· Since A compact, {xn} has a
subsequence converging to a point of A ; say Xnk --> x E A. Similarly {Ynk} has
a subsequence Yn~ --> y E A. Since { n~} is a subsequence of { nk}, Xn~ --> x.
Then bn' k = lxn' k - Yn' k I --> Ix - YI E B. :. B is compact.
By Ex. 9, supB EB.:. 3 xo,Yo EA 3 lxo-Yol = sup{lx-yl: x,y EA}=
d(A).