 # 1. (20) For each of the following subsets of ℝ, give its maximum,...

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1. (20) For each of the following subsets of ℝ, give its maximum, minimum, infimum, and supremum, if they exist. Otherwise, write "none". (a) {1,4} (b) [0,4] (c) { } (d) { } (e){ ( ) } (f) ⋂ ( ) (g) ⋃ [ ] (h) (i) (j) {∑ 2. (10) Prove that every sequence of real numbers has a monotone subsequence. 3. (10) The sequence given by converges. Find its limit. 4. (10) Prove √ . 5. (10) Prove that if , then √ √ for all . 6. (10) For any sequence , let (a) Prove that implies . (b) Give an example of a divergent sequence such that converges. 7. (10) Suppose that (an) is a sequence of numbers such that for all n, | | where ∑ is convergent. Show that (an) converges. 8. (20) Prove the following, using Definition 2.2.3(convergence of a sequence): (a) (b)

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