practicexam1-08

practicexam1-08 - b n ) is bounded as well. 5)True or False...

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18.100B/C FALL 2008: PRACTICE EXAM #1 Problems 1) Let ( X,d ) be a metric space, x 0 X and r > 0. Let B = B ( x 0 ,r ) = { x X/d ( x 0 ,x ) < r } and C = { x X/d ( x 0 ,x ) r } . a) Prove that ¯ B C . b) Give an example of a space ( X,d ), a point x 0 and a radius r for which ¯ B 6 = C . c) Prove that in R n , with the Euclidean metric, ¯ B = C. 2) Assume ( a n ) , ( b n ) and ( c n ) are sequences in R . Assume also that for all n n 0 we have a n b n c n . Assume also that lim n →∞ a n = lim n →∞ c n = l, where l is a real number. Prove that lim n →∞ b n = l. 3) Let A and B be countable sets. Prove that A B is countable and that A B is at most countable using the definition of countability. 4) Assume that ( a n ) and ( b n ) are two sequences or rational numbers that are equivalent in the sense discussed in class. Assume also that ( a n ) is a bounded sequence. Prove that (
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Unformatted text preview: b n ) is bounded as well. 5)True or False (justify your answer!): a) For any open set A R , we have int ( A ) = A . b) Let V be the space of all functions f : [0 , 1] R , and dene k f k = | f (0) | . Is ( V, k k ) a eld with a norm, if we use the usual sum and multiplication of real valued functions? 6) Show by induction that r 2 + q 2 + 2 + ... | {z } n roots = 2 cos 2 n +1 . Then use this equality to nd the limit of the recursive sequence given by a 1 = 2 a n = 2 + a n n > 1 . 7) Give an example of a countable compact subset of ( R ,d ), where d is the Euclidean metric. 1...
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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.

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