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Unformatted text preview: Math 5615H. November 16, 2011. Midterm Exam 2. Problems and Solutions. Problem 1. (10 points). Let A 1 ,A 2 ,A 3 ,... be subsets of a metric space ( X,d ). If B := ∞ [ k =1 A k , prove that B ⊇ ∞ [ k =1 A k , where B and A k denote the closures of B and A k . Show, by an example, that this inclusion can be proper. Proof. We have for all k = 1 , 2 , 3 ,... B ⊇ A k = ⇒ B ⊇ A k = ⇒ B ⊇ ∞ [ k =1 A k . On the other hand, consider the countable set Q = { q 1 ,q 2 ,q 3 ,... } of all rational numbers in R . For k = 1 , 2 , 3 ,... , take A k := { q k } – the set consisting of a single point q k . Then A k = A k , B := ∞ [ k =1 A k = Q, B = Q = R % ∞ [ k =1 A k = ∞ [ k =1 A k = Q. Problem 2. (12 points). The sequence { a n } is defined by a 1 = 0 , and a n +1 = 2 a n / 2 for n = 1 , 2 , 3 ,.... Prove that { a n } is convergent and find its limit. You can use without proof the fact that the function y = f ( x ) = a x , where a > , a 6 = 1, is strictly convex , i.e....
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This document was uploaded on 02/15/2012.
 Fall '09
 Sets

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