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# hw2 - Ç-N proof that lim n â†’âˆž 1 n 1 = 0 Note This is...

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MATH 410: Homework 2 Due in class Friday 2/10/2012 Section 1.2 1. For each of the following statements, determine whether it is true or false and informally justify your answer. (a) Z is dense in R . * (b) R > 0 is dense in R . * (c) Q \ Z is dense in R . * Note: This is book problem 1 with a misprint fixed. 2. Suppose that S Z is nonempty and bounded below. Show S has a minimum. * Note: This is part of book problem 2. Modify the proof of book Proposition 1.7. 3. Show that c R , ! x Z with x ( c, c + 1]. ** Note: This is book problem 7. Don’t forget the uniqueness. Section 1.3 1. Let a, b R with | a b | ≤ 1. Prove that | a | ≤ | b | + 1. * Note: This is book problem 8. You can assume (and use and reference) book problem 7 if you wish. 2. For n N and a, b R with a b 0 prove that a n b n nb n - 1 ( a b ). ** Note: This is book problem 10. 3. Show that if a R , a negationslash = 0 that 1 a = 1 + (1 a ) + (1 a ) 2 + (1 - a ) 3 a . * Note: This is book problem 20b. Do not use brute force, intead apply the Geometric Sum Formula with an appropriate r and n then do very little algebra. Section 2.1 1. Using only the AP give a direct
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Unformatted text preview: Ç«-N proof that lim n â†’âˆž 1 n +1 = 0. * Note: This is book problem 2b. 2. Using only the AP give a direct Ç«-N veriÂ±cation that b 2 âˆš n + 1 n + 3 B converges. ** Note: This is book problem 3a. 3. For the sequence { a n } deÂ±ned in book Example 2.3 show that âˆ€ x âˆˆ Q âˆ© (0 , 1] there are inÂ±nitely ** many indices n such that a n = x . ClariÂ±y what this is saying with a speciÂ±c example. Note: This is book problem 5 plus a bit more. 4. Suppose that { a n } converges to a > 0. Show âˆƒ N st n â‰¥ N â†’ a n > 0. * Note: This is book problem 6. 5. DeÂ±ne a 1 = 1 and a n +1 = Â± a n + 1 n if a 2 n â‰¤ 2 a n âˆ’ 1 n if a 2 n > 2 (a) Write the Â±rst Â±ve terms of this sequence. * (b) Show that âˆ€ n , | a n âˆ’ âˆš 2 | < 2 n . ** (c) Use this property to show the sequence converges to âˆš 2. * Note: This is book problem 12 broken up plus a bit more....
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