Question

3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover. 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact. 5-2) Now use induction to show that a finite union of compact subsets of R is compact. 5-3) Let A be a nonempty finite subset of R. Prove that A is compact. 5-4) Give an example of a countable union of compact sets that is not compact. (Hint: Recall that every compact subset of R is closed and bounded. The set N is not bounded and so it is not compact. However, N = = {n}. Note that it follows from what you proved in the previous item that for each n e N, the singleton {n} is compact)

Please help me solve 3,4,5

Answer 1

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