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midterm2-315-f2006-solutions

midterm2-315-f2006-solutions - Math 315 Section 1 Fall 2006...

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Math 315 Section 1 — Fall 2006 — Midterm Exam 2 — Solutions 1. Give a simple example of each of the following, or argue that such a request is impossible. (a) A Cauchy sequence with a divergent subsequence. This is impossible. Every subsequence of a convergent sequence (i.e. a Cauchy sequence) converges to the same limit as the original sequence. [See Theorem 2.5.1] (b) An unbounded sequence containing a subsequence that is Cauchy. Let a n = 1 /n is n is odd and a n = n is n is even. That is, ( a 1 , a 2 , a 3 , a 4 , . . . ) = ( 1 1 , 2 , 1 3 , 4 , 1 5 , 6 , . . . ) . This sequence is unbounded because the subsequence ( a 2 k ) is unbounded, but the subse- quence ( a 1 , a 3 , a 5 , a 7 , . . . ) is a Cauchy sequence. (c) A continuous function f : (0 , 1) R and a Cauchy sequence ( x n ) such that ( f ( x n ) ) is not a Cauchy sequence. Let f ( x ) = 1 /x and ( x n ) = (1 /n ) . Then f ( x ) is continuous on (0 , 1) and the sequence ( x n ) is Cauchy, but f ( x n ) = n. So, the sequence ( f ( x n ) ) is not a Cauchy sequence. (d) A continuous function f : [0 , 1] R and a Cauchy sequence ( x n ) such that ( f ( x n ) ) is not a Cauchy sequence. This is impossible. Continuous functions on closed sets map Cauchy sequences to Cauchy sequences. [See Theorem 4.3.2(iv).] (e) A function f that is differentiable on the interval [0 , 1] such that f (0) = - 1 and f (1) = 1, but f ( x ) is never equal to 0 for any x [0 , 1]. This is impossible. By Darboux’s Theorem, since f (0) = - 1 < 0 < f (1) = 1 , there must exist some c (0 , 1) where f ( c ) = 0 . 1
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2. For each of the following statements, circle True or False. No justification is necessary. For any set A R , ( ¯ A ) c is open. True False A set A is closed if and only if A = ¯ A . True False If A is a bounded set, then s = sup A is a limit point of A . True False An open set that contains every rational number must necessarily be all of R . True False An arbitrary intersection of compact sets is compact. True False If F 1 F 2 F 3 ⊇ · · · is a nested sequence of nonempty closed sets, then the intersection n =1 F n = .
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