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Unformatted text preview: 1.112 We proceed sequentially as follows. Choose any 1 in . If the open ball ( 1 , ) contains , we are done. Otherwise, choose some 2 / ( 1 , ) and consider the set 2 =1 ( , ). If this set contains , we are done. Otherwise, choose some 3 / 2 =1 ( , ) and consider 3 =1 ( , ) 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 ( 1 , 2 , 3 ,... ) which had no convergent subsequence. The would contradict the compactness of . 1.113 Assume is compact. The previous exercise showed that is totally bounded. Further, since every sequence has a convergent subsequence, every Cauchy sequence converges (Exercise 1.111). Therefore is complete....
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.
- Fall '10