1549901369-Elements_of_Real_Analysis__Denlinger_

3.4 *The Cantor Set 171 where each Cn is the union of 2n disjoint closed intervals, all of length _2_. 3n Let € > 0. Each of ...

1 72 Chapter 3 • Topology of the Real Number System (μ5) If A and Bare measurable, then μ(A - B) =μ(A) - μ(An B), and μ(AU B) =μ ...

3.4 *The Cantor Set 173 2 Then, let R2 = the union of the 2 open intervals of length r 2 , one centered in each of the 2 closed ...

174 Chapter 3 • Topology of the Real Number System \:/n E .N, let Ln denote the set of all left endpoints of the disjoint close ...

3.4 *The Cantor Set 175 (Project) Fat Cantor-like Sets: For "fat" Cantor-like sets C(a) de- fined in Definition 3.4.23, which ...

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Chapter 4 Limits of Functions Limits are basic to analysis. But we have done much of the hard work in Chapter 2. In this chapter ...

178 Chapter 4 • Limits of Functions off could not get "close to" x 0. Now, saying that f(x) gets close to Lis saying that lf(x) ...

4.1 Definition of Limit for Functions 179 (4) If Xo is an interior point of V(f), then Definition 4.1.l simplifies to: lim f(x) ...

180 Chapter 4 • Limits of Functions Example 4.1.2 Consider the limit statement lim(4x - 5) = 3. X->2 (a) Find a value of o &g ...

4.1 Definition of Limit for Functions 181 Example 4.1.3 Prove that lim(4x - 5) = 3. X->2 (This example should remind you of E ...

182 Chapter 4 • Limits of Functions Solution: (a) We want a real number o > 0 such that 0 < Ix - 21 < o::::} lx^2 - 41 ...

4.1 Definition of Limit for Functions 183 Example 4.1.6 Prove that lim x^2 = 4. X->2 Proof. Let c. > 0. Choose o = min{l, ...

184 Chapter 4 • Limits of Functions Proof. Apply the rules of quantifier negation^3 to Definition 4.1.1. • Theorem 4.1.8 (Unique ...

4.1 Definition of Limit for Functions 185 'Vo> 0, 3x E V(f) 3 0 < Ix - xol < o but lf(x) - LI ~ c. Keep this c fixed th ...

186 Chapter 4 • Limits of Functions Proof. Let f ( x) = sin ( ~). Consider the sequences { Xn} = { nl7f } and {yn} = { 1 }. Then ...

4.2 Algebra of Limits of Functions 187 Prove each of the following limit statements using Definition 4.1.1: (a) lim (2x + 13) = ...

188 Chapter 4 • Limits of Functions we look for the implications of 0 < Ix - xol < 8 instead of n 2: no. As you go through ...

4.2 Algebra of Limits of Functions 189 t t t x 0 -e Xo x 0 +e Figure 4.3 The language of neighborhoods and deleted neighborhoods ...

190 Chapter 4 • Limits of Functions Definition 4.2.8 A function f is said to be bounded away from^7 0 on a set A~ V(f) if :JC> ...