1549901369-Elements_of_Real_Analysis__Denlinger_

3.2 Closed Sets and Cluster Points 151 (b) (-¢=): Suppose :3 sequence {an} of points of A other than x, such that an ----; x. Th ...

152 Chapter 3 • Topology of the Real Number System Tell whether the following sets are open, closed, both, or neither: (a) (3, ...

3.2 Closed Sets and Cluster Points 153 Find the closure of each of the following sets: (a) (3, 5) U {6} (c) {1,2,3,4,5,6, 7,8, ...

154 Chapter 3 • Topology of the Real Number System (Project) In this exercise we shall denote the closure of a set A by Ac! rat ...

3.3 * Compact Sets 155 Notice that the following are not topological terms: "interval,'' "bounded,'' "sup A,'' "inf A,'' "Archim ...

156 Chapter 3 • Topology of the Real Number System Let n be a fixed natural number. Then the sets N :!;. ( ~) = ( ~^1 , k~^1 ) f ...

3.3 * Compact Sets 157 I Thus, U is an open cover of (0, 1). Clearly, no finite subcollection of U will cover (0, ]). Thus, U ha ...

158 Chapter 3 11 Topology of the Real Number System Corollary 3.3.9 (a) JR (b) (a, b) ( c) (a, b] (d) [a, b) (e) (-oo,a) (f) (-o ...

3.3 *Compact Sets 159 Claim #2: u = b. Proof: For contradiction, suppose u "I-b. Then u < b, since a ~ u ~ b. Since u E S, so ...

160 Chapter 3 • Topology of the Real Number System Theorem 3.3.13 (Sequential Criterion for Compactness) A set A of real numbers ...

3.3 * Compact Sets 161 In either case, { Xn} has a subsequence converging to a point of A. Therefore, by Theorem 3.3.13, A is co ...

162 Chapter 3 111 Topology of the Real Number System Theorem 3.3.19 A set A of real numbers is compact if and only if, for every ...

3.3 * Compact Sets 163 By our hypothesis, n C contains a point of A. But, n C = n {Uc: U EU}= (U Vt. Thus, (u Vt contains a poin ...

164 Chapter 3 • Topology of the Real Number System Let f : V(f) --+ JR be a function with domain V(f), and let A ~ V(f). 3.4 W ...

3.4 *The Cantor Set 165 Thus, C2 = [O, i] urn,~] u [~, ~] u [~, 1]. Continuing inductively, if Cn is the union of 2n disjoint cl ...

166 Chapter 3 • Topology of the Real Number System E ach base-b decimal represents a unique real number. As in Section 2.5, Vn E ...

3.4 *The Cantor Set 167 (d) 0.1111111 · · · (bas; 3) = i + i + 217 + · · · 1 1 ~ 3 ~ ~~ - · 1-i 3-1 2' ( Recall t: geometri~ s ...

168 Chapter 3 • Topology of the Real Number System of some number in [O, 1] since it consists of all O's and l's. Moreover, ther ...

3.4 *The Cantor Set 169 Since 3 : < c, either an E Nc;(x) or bn E Nc;(x). Recall that both an and bn are in the Cantor set. H ...

170 Chapter 3 • Topology of the Real Number System SETS OF MEASURE ZERO Is the Cantor set a large or small subset of [O, l]? In ...