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19. Give an example of a set with a countably infinite set of accumulation points. 20 Give on pints

19. Give an example of a set with a countably infinite set of accumulation points. 20 Give on pints

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 19 and 22 please? Thanks

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). *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 23. Let ao and a, be distinct real numbers. Define an for each positive integer 2 1 n≥ 2. Show that {an) is a Cauchy sequence. You may want to use induction to show that an+1 = an = n (-1)^* (a₁ - ao) and then use the result from Example 0.9 of Chapter 0.
Transcribed Image Text: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). *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 23. Let ao and a, be distinct real numbers. Define an for each positive integer 2 1 n≥ 2. Show that {an) is a Cauchy sequence. You may want to use induction to show that an+1 = an = n (-1)^* (a₁ - ao) and then use the result from Example 0.9 of Chapter 0.
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