1549901369-Elements_of_Real_Analysis__Denlinger_

2.9 *Upper and Lower Limits 131 is called the degree of x. For example, J3 is algebraic of degree 2, since it satisfies x^2 - 3 ...

132 Chapter 2 • Sequences Definition 2.9.2 (Lower Limit) Suppose {xn} is any sequence ofreal num- bers. Case 1: If { Xn} is boun ...

2.9 *Upper and Lower Limits 133 (d) Vn E N, Wn = - oo, so lim Wn = - oo. Also, Vn E N, Wn = -n, so n-+oo n-+oo lim Wn = - oo. D ...

134 Chapter 2 • Sequences Therefore, since limits preserve inequalities, lim Xnm ~ lim Xm and m-+oo m-+oo lim Xnm ::::; lim Xm. ...

2.9 *Upper and Lower Limits 135 Theorem 2.9.9 A bounded sequence { Xn} converges if and only if lim Xn and n->oo lim Xn are b ...

136 Chapter 2 • Sequences For each of the following, prove or find a counterexample for which the given equation is not true: ( ...

Chapter 3 Topology of the Real Number System Sections 3. 1 and 3.2 present the concepts of neighborhoods, open and closed sets, ...

138 Chapter 3 • Topology of the Real Number System analysis. Indeed, every aspiring mathematician should eventually take a cours ...

3.1 Neighborhoods and Open Sets 139 Definition 3.1.3 A set U ~ JR is open if Vx E U, ::le > 0 3 N,:(x) ~ U. In words, a set U ...

140 Chapter 3 • Topology of the Real Number System (b) Let C be any collection of open sets. To prove that UC is open, let x E U ...

3.1 Neighborhoods and Open Sets 141 INTERIOR, EXTERIOR, AND BOUNDARY Definition 3.1.9 (Interior of a Set) Let A be a set of real ...

142 Chapter 3 m Topology of the Real Number System Definition 3.1.12 (Exterior of a Set) Let A be a set of real numbers. The ext ...

3.1 Neighborhoods and Open Sets 143 Examples 3.1.16 Some boundary points: (a) 3 and 6 are boundary points of the intervals (3, 6 ...

144 Chapter 3 • Topology of the Real Number System Proof. Exercise 15. • Theorem 3.1.22 (Finite Sets) (a) Finite sets have no in ...

3.1 Neighborhoods and Open Sets 145 Suppose A is a bounded, nonempty set of real numbers. Prove that sup A and inf A are bounda ...

146 Chapter 3 • Topology of the Real Number System 3.2 Closed Sets and Cluster Points The notion of "closed set" is closely rela ...

3.2 Closed Sets and Cluster Points 147 The closed set theorem claims that the intersection of any collection of closed sets is c ...

148 Chapter 3 • Topology of the Real Number System Hence, :3 x E Ac 3 x is a cluster point of A. That is, there is a cluster poi ...

3.2 Closed Sets and Cluster Points 149 Theorem 3.2.13 (Balzano-Weierstrass Theorem for Sets) Every bounded infinite set of real ...

150 Chapter 3 111 Topology of the Real Number System Examples 3.2.16 Let A= [O, 1), B = [O, 1) U {2}, and C = (0, 3) U (3, 5). ( ...