1549901369-Elements_of_Real_Analysis__Denlinger_

2.2 Algebra of Limits 71 Suppose lim Xn = S and lim Yn = -3. Find each of the following: n-+oo n-+oo x2 - 2y2 <(a) lim (2x ...

72 Chapter 2 • Sequences '17. Prove that each of the following sequences diverges: (a) {4n-5} (b) { J3n + 1} ( c) { 2~ 3 } ( d) ...

2.3 Inequalities and Limits 73 Outline of a proof of (a) [fill in missing steps, and justify]: c:. Case 1 (L = 0): Then 3n 1 EN ...

7 4 Chapter 2 • Sequences Let no= max{n1,n2}. Then n 2: no ==? n 2: n1 and n 2: n2 =} Ian - LI < € and lcn - LI < € =} -€ ...

2.3 Inequalities and Limits 75 Now~ -t 0. Therefore, by the second squeeze principle, with an= 2 n +^3 n 3n - 7 ' 2 3 2n + 3 2 L ...

76 Chapter 2 • Sequences Moreover, \In EN, an :'.": u =inf A. Thus, {an} is a sequence of elements of A such that \In E N, 1 1 u ...

2.3 Inequalities and Limits 77 Example 2.3.8 lim efFi, = l. (Assume the existence^9 of efri,.) n-+oo 'f1Yn Proof. Vn E N, let an ...

78 Chapter 2 • Sequences *Theorem 2.3.10 If {xn} is a sequence of nonzero numbers such that lim IXn+l I = L < 1, then, Xn --- ...

2.3 Inequalities and Limits 79 LIMITS PRESERVE INEQUALITIES In our future work we shall have many occasions to use the following ...

80 Chapter 2 • Sequences EXERCISE SET 2.3 Prove Corollary 2.3.2. In each of the following, find the limit and use the second sq ...

2.3 Inequalities and Limits 81 ( c) Calculate hm. (9 - + -^9 + --^9 + · · · + -9). What does this tell n-><X> 10 100 10 ...

82 Chapter 2 • Sequences Prove that if all terms of a convergent sequence { Xn} lie in a closed interval [a, b], then its limit ...

2.4 Divergence to Infinity 83 Thus, we want to find an no E N 3 3n- 2 n 2:: no :::;, n 2:: 24 and - 6 100; i.e., n 2:: no ...

84 Chapter 2 • Sequences Therefore, lim ( 3 n 2 -;;) = +oo. O n-+oo 5n + Theorem 2.4.4 If { Xn} is a sequence of positive real n ...

2.4 Divergence to Infinity 85 Proof. By Example 2.4.5 above, lim (~)n = +oo. Now, 'tin EN, n--+oo 3 2n +4n Therefore, by Theorem ...

86 Chapter 2 • Sequences n > no :::::} an > M and Cn < -1 :::::} an > M and - Cn > 1 :::::} (an)(-Cn) > M :::: ...

2.4 Divergence to Infinity 87 We can improve our statement about the indeterminate ~ by introducing a further bit of notation. S ...

88 Chapter 2 11 Sequences Use Definition 2.4.1 to prove each of the limit statements in Exercise 1. Prove each of the following ...

2.5 Monotone Sequences 89 Limit Comparison Test: Suppose {xn} and {Yn} are sequences of positive real numbers such that Xn ---+ ...

90 Chapter 2 • Sequences (e) monotone if it is any one of (a) or (b) or (c) or (d). (f) strictly monotone if it is either (c) or ...