Unformatted text preview: Suppose S is a nonempty subset of R . Prove that there exists a sequence ( s n ) such that (i) s n ∈ S for any n ∈ N , and (ii) lim s n = sup S . Hint . Prove that, for every n ∈ N , S ∩ (sup S − 1 /n, sup S ] n = ∅ . Use this fact to select s n . 7. Determine whether the following sequences converge. Compute the limits if they exist. (a) a n = √ n 4 + 4 n − n 2 , (b) b n = (2 n + cos n ) / ( n + 1), (c) c n = ( − 1) n n 2 . 8. Prove that ( √ 5 − 1) / √ 2 is not a rational number. 9. Let a n = n/ 2 n . Prove that lim n →∞ a n = 0. Hint . You can use Exercise 1.9 (from Homework 1). 10. ±ind all rational numbers r satisfying 2 r 5 − 8 r 2 + 1 = 0. Links: the syllabus, the main page of the course. the solutions. 1...
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 Winter '08
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 Addition, Prime number, Prove

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