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Macroeconomics Exam Review 13

Macroeconomics Exam Review 13 - Solutions for Foundations...

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1.112 We proceed sequentially as follows. Choose any 𝑥 1 in 𝑋 . If the open ball 𝐵 ( 𝑥 1 , 𝑟 ) contains 𝑋 , we are done. Otherwise, choose some 𝑥 2 / 𝐵 ( 𝑥 1 , 𝑟 ) and consider the set 2 𝑖 =1 𝐵 ( 𝑥 𝑖 , 𝑟 ). If this set contains 𝑋 , we are done. Otherwise, choose some 𝑥 3 / 2 𝑖 =1 𝐵 ( 𝑥 𝑖 , 𝑟 ) and consider 3 𝑖 =1 𝐵 ( 𝑥 𝑖 , 𝑟 ) The process must terminate with a finite number of open balls. Otherwise, if the process could be continued indefinitely, we could construct an infinite sequence ( 𝑥 1 , 𝑥 2 , 𝑥 3 , . . . ) which had no convergent subsequence. The would contradict the compactness of 𝑋 . 1.113 Assume 𝑋 is compact. The previous exercise showed that 𝑋 is totally bounded. Further, since every sequence has a convergent subsequence, every Cauchy sequence converges (Exercise 1.111). Therefore 𝑋 is complete. Conversely, assume that 𝑋 is complete and totally bounded and let 𝑆 1 = { 𝑥 1 1 , 𝑥 2 1 , 𝑥 3 1 , . . . } be an infinite sequence of points in 𝑋 . Since
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