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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 2-108. 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
- Equivalence relation, Rational number, equivalence class, Cauchy sequence