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Unformatted text preview: Homework answers, week of Sept. 2 Practice problems 1. Use an inductive argument as outlined in the Aug. 28 lecture. 2. Use l’Hˆopital’s rule: (a) 1, (b) 0. 3. There are many possibilities. The following sequence has accumulation points at 0, 1 / 2, and 1: . 1 , . 4 , . 9 , . 01 , . 49 , . 99 , . 001 , . 499 , . 999 ,... Homework problems 1. (a) (i) I 1 = ( 1 2 , 3 2 ) , I 2 = ( , 1 ) , I 3 = ( 1 6 , 5 6 ) , I 4 = ( 1 4 , 3 4 ) ,. . . . (ii) I 1 = ( , 1 ) , I 2 = ( , 1 2 ) , I 3 = ( , 1 3 ) , I 4 = ( , 1 4 ) , . . . . (b) (i) T ∞ n = 1 I n = 1 2 . (ii) T ∞ n = 1 I n = /0 (the empty set). 2. Sequence (ii) is a sequence of nested open intervals whose intersection is empty, because 1 / n → 0, yet 0 does not belong to any of the intervals. Consequently, this sequence of intervals is shows that the statement of the nested intervals theorem is false if the hypothesis of closed intervals is re placed with open ones. (Remember: one counterexample suffices to prove a statement false.)a statement false....
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 Spring '07
 thieme
 Logic, Inductive Reasoning, Empty set, Supremum, sup, 0.999...

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