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Unformatted text preview: 18.100B/C: Fall 2008 Homework 4 Available Tuesday, September 30 Not due If you would like feedback on your solutions, you can turn in the homework by 11am on Wednesday, October 8, in 2108. For 18.100B it should be put in the bin corresponding to the lecture you regularly attend. For 18.100C always put it in the C bin. We write ( a n ) ∞ n =1 or ( a n ) n ∈ N or simply ( a n ) for sequences. 1. Let ( a n ) ∞ n =1 be a sequence in R with the property that no subsequence converges. Prove that  a n  → ∞ . Does the same property hold if the a n are in Q and we consider ( a n ) ∞ n =1 as a sequence in the metric space Q ? 2. (a) Let ( a n ) be a Cauchy sequence in Q such that a n does not converge to . Show that there is an ω > and an N ∈ N so that  a n  > ω for all n ≥ N . Moreover, show that the sign of a n is constant for large n . (b) Following part (a) , show that if ( a n ) is a Cauchy sequence in Q that does not con verge to , then (1 /a n ) is a Cauchy sequence in...
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 Fall '10
 Prof.KatrinWehrheim
 Equivalence relation, Rational number, equivalence class, Cauchy sequence

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