This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View Full
Document
 Fall '09
 Sets

Click to edit the document details