1549901369-Elements_of_Real_Analysis__Denlinger_

2.1 Basic Concepts: Convergence and Limits 51 or as a vector with infinitely many components, {xn} = (x1, Xz, X3,... 'Xn, Xn+l, ...

52 Chapter 2 • Sequences We can also plot the terms of the sequence as points on a number line, as in Figure 2.1 (b), and notice ...

2.1 Basic Concepts: Convergence and Limits 53 is infinitely large or infinitely small, we must find a way to express the concept ...

54 Chapter 2 • Sequences 1 Solution: (a) We want an no EN 3 n 2'. no::::} 2n+^3 21 3 n _ 7 3 < .01. Now, I 2n + 3 _ ~I = I ...

2.1 Basic Concepts: Convergence and Limits 55 The latter inequality will be true if 9n - 21 23 > 2,000 9n - 21 > 46,000 9n ...

56 Chapter 2 • Sequences . 3 By the Arch1medean property (remember that?), :3no EN 3 no> - + 3. € Take no= such a natural num ...

2.1 Basic Concepts: Convergence and Limits 57 I Now, n > 3 ::::}^23 I^23 9 n _ 21 = 9 n _ 21 . Thus, n ::'.". no ::::} I 23 I ...

58 Chapter 2 • Sequences I 3n 2 Solution: (a) We want an n - 4n I 0 EN^3 n ~no=? n 2 + 5 - 3 < .01. Now, I 3n 2 4n _ 31 = I ...

2.1 Basic Concepts: Convergence and Limits 59 The latter inequality will be true if n^2 + 5 4 n + 15 > 2, 000; i.e., n^2 + 5 ...

60 Chapter 2 • Sequences }-Let no be such a natural number. The above analysis shows that n ~no=? 1 3 ~: ~ ~n - 31 < ~ < € ...

2.1 Basic Concepts: Convergence and Limits 61 SUMMARY: HOW TO PROVE lim xn = L n-+oo Let c > 0. Find a real number r such th ...

62 Chapter 2 • Sequences --(a) (c) __.....(e) --(g) ./ (i) lim --; = 0 n--tex> n 7n lim --=0 n-+= n^2 + 3 lim ~ =3 n-+= n + ...

2.2 Algebra of Limits 63 Theorem 2.2.3 A constant sequence {xn} = {c} converges (to c). Proof. Exercise 3. • Definition 2.2.4 A ...

64 Chapter 2 11 Sequences Let n 0 = max{!IJ._, n2l· Then no ?: n 1 and no ?: n2, and so e e Ix ~ -L/ < - 2 and /x __?!-O. -Ml ...

2.2 Algebra of Limits 65 Proof. Let { Xn} denote a convergent sequence, say Xn -t L. Taking t: = 1 in Definition 2.1.4, 3 no E N ...

66 Chapter 2 • Sequences lfiJVM Theorem 2.2.13 (Algebra of Limits) Suppose {xn} and {yn} are conver- ,,, ,,, gent sequences and ...

2.2 Algebra of Limits 67 Let no= max{n 1 ,n2}. Then n 2:: no =? n 2:: n1 and n 2:: n2 c c ::::} lxn - LI < 2 and IYn -Ml < ...

68 Chapter 2 11 Sequences =} JXnYn - LMJ < €. Therefore, lim (XnYn) = LM = lim Xn· lim Yn· n-?OO n-+oo n-+oo (e) Let € > 0 ...

2.2 Algebra of Limits 69 Case 2 (L > 0): Since Xn---+ L , 3 n2 EN 3 n :'.'.'. n2::::} lxn - LI < c.JL. Let no= max{n1,n2}. ...

70 Chapter 2 • Sequences We have already had occasion to make use of this fact. An "eventually constant" sequence (Definition 2. ...