1549901369-Elements_of_Real_Analysis__Denlinger_

2.6 Subsequences and Cluster Points 111 Proof. Exercise 7. • Theorem 2.6.13 L et {xn} be a sequence. Then (a) {xn} diverges to + ...

112 Chapter 2 • Sequences Thus, we see that the cluster points are 0, 1, and +oo. D BOLZANO-WEIERSTRASS THEOREM The following th ...

2.6 Subsequences and Cluster Points 113 Theorem 2.6.17 A bounded sequence converges {::} it has one and only one cluster point. ...

114 Chapter 2 • Sequences Define Ek+i = L - Xnk. Then Ek+i > 0, so by (13), :J infinitely many n EN 3 Xn E (L -Ek+1, L). Henc ...

10. 2.6 Subsequences and Cluster Points 115 Find all cluster points of the following sequences. Then use Corollary 2.6.10 or The ...

116 Chapter 2 • Sequences Suppose {xn} is a bounded sequence. Prove that if all its convergent subsequences have the same limit ...

·'!'" 2.7 Cauchy Sequences 117 First show^14 that Vu, v E JR, ..jUV :::; u ; v and equality holds <¢=> u = v. Use this to ...

118 Chapter 2 • Sequences Proof. Suppose { Xn} is a Cauchy sequence. Then, taking E = 1 in Defini- tion 2.7.1, :lno EN 3 m,n ~no ...

2.7 Cauchy Sequences 119 c Proof. Suppose that {xn} is a sequence 3 \fn E N, lxn+l - Xnl < 2n Then, whenever m > n in N, l ...

120 Chapter 2 11 Sequences Suppose that {xn} is a sequence such that Vn EN, [xn+1 -xn[ <en, for some constant 0 < C < ...

2.7 Cauchy Sequences 121 10. (Project) Recursive Weighted Arithmetic Means: Let a -=j; b be arbitrary real numbers, let 0 < t ...

122 Chapter 2 • Sequences (g) (h) an - (Jn Vn EN, define Un = , where a and (3 are as defined in (f). a -(3 Prove that u 1 = 1, ...

2.7 Cauchy Sequences 123 *PROPERTIES EQUIVALENT TO COMPLETENESS Several of the major theorems we have encountered so far are equ ...

124 Chapter 2 • Sequences X2 is an upper bound for S; X2 '.S X1i 1 lx2 - xii~ 2; x 2 - ~ is not an upper bound for S, because, ...

2.8 *countable and Uncountable Sets 125 Step 2. By (15) above, and by Theorem 2.7.5 above, {xn} is a Cauchy sequence. By our hyp ...

126 Chapter 2 • Sequences in 1-1 correspondence with N , and is uncountable otherwise. To make these ideas clear we must begin w ...

2.8 *countable and Uncountable Sets 127 Theorem 2.8.5 There is a sequence whose range is Q. That is, the set of rational numbers ...

128 Chapter 2 • Sequences rational numbers are listed more than once in the sequence f. By eliminating the repetitions, we obtai ...

2.8 *countable and Uncountable Sets 129 y =f. Xm, since y and Xm differ in the mth decimal place and em =f. 0, 9; Thus, we have ...

130 Chapter 2 • Sequences Prove that the relation ~ of Definition 2.8. l has the following properties: (a) (Reflexivity) \IA, ...