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lan 16. Prove directly (do not use Theorem 1.9) that, if (a) and (bn)n-1 are Cauchy, so (a,bn)-1. You will want to use Theorem 1.4. 2n+1 17. Prove that the sequence n 00 n=1 is Cauchy.

lan 16. Prove directly (do not use Theorem 1.9) that, if (a) and (bn)n-1 are Cauchy, so (a,bn)-1. You will want to use Theorem 1.4. 2n+1 17. Prove that the sequence n 00 n=1 is Cauchy.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Could you do 16 and 17 please? 
for 16 theorem 1.4 is "every Cauchy sequence is bounded"

{an + bn}=1. 16. Prove directly (do not use Theorem 1.9) that, if {a}=1 and (bn)-1 are Cauchy, so is {ambnl=1. You will want to use Theorem 1.4. 2n+1 17. Prove that the sequence n n=1 18. Give an example of a set with exactly two accumulation points. 19. Give an example of a set with a countably infinite set of accumulation points. 20. Give an example of a set that contains each of its accumulation points. 21. Determine the accumulation points of the set {2"+:n and k are positive integers). k 22. Let S be a nonempty set of real numbers that is bounded from above (below) and let x = sup S (inf S). Prove that either x belongs to S or x is an accumulation point of S. an-1 + An-2 for each positive integer 2 8 3. Let a, and a, be distinct real numbers. Define an = is Cauchy. ∞ an+1 = an n≥ 2. Show that {an) is a Cauchy sequence. You may want to use induction to show that = n (-3)*( and then use the result from Example 0.9 of Chapter 0. (a₁ – ao) 1
Transcribed Image Text:{an + bn}=1. 16. Prove directly (do not use Theorem 1.9) that, if {a}=1 and (bn)-1 are Cauchy, so is {ambnl=1. You will want to use Theorem 1.4. 2n+1 17. Prove that the sequence n n=1 18. Give an example of a set with exactly two accumulation points. 19. Give an example of a set with a countably infinite set of accumulation points. 20. Give an example of a set that contains each of its accumulation points. 21. Determine the accumulation points of the set {2"+:n and k are positive integers). k 22. Let S be a nonempty set of real numbers that is bounded from above (below) and let x = sup S (inf S). Prove that either x belongs to S or x is an accumulation point of S. an-1 + An-2 for each positive integer 2 8 3. Let a, and a, be distinct real numbers. Define an = is Cauchy. ∞ an+1 = an n≥ 2. Show that {an) is a Cauchy sequence. You may want to use induction to show that = n (-3)*( and then use the result from Example 0.9 of Chapter 0. (a₁ – ao) 1
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