104_Fa08_MT1_sol - Midterm 1 for MATH 104 , Fall 2008...

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Unformatted text preview: Midterm 1 for MATH 104 , Fall 2008 Thursday, September 25 Problem 1 [5P] State the definition of a Cauchy sequence in R precisely. Problem 2 [9P] For each of the following sets in R , determine whether it is bounded from above, and if so, determine its least upper bound. (a) { (- 1) n- 1 n : n N } Solution. Least upper bound is 1. (b) { n 41 + 1 n : n N } Solution. Least upper bound is 42. (c) { q Q : q is the di ff erence of two irrationals } Solution. The set contains every rational number r , since if x = 2, y = 2 + r , then x , y irrational and y- x = r . Problem 3 [9P] Show that if A is a non-empty and bounded subset of R , then there exists a sequence ( a n ) such that for all n N , a n A , and such that lim n a n = l . u . b . ( A ). Solution. Let b = sup( A ). We know that for every n N , there exists an s n A such that b- 1 / n s n b . We claim that s n b . Let > 0. Since 1 / n 0, there exists N such that for all n...
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104_Fa08_MT1_sol - Midterm 1 for MATH 104 , Fall 2008...

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