Macroeconomics Exam Review 15

Macroeconomics Exam Review 15 - Choose any =1 . That is,...

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Unformatted text preview: Choose any =1 . That is, for each = 1 , 2 ,... . Thus, for every > 0 and = 1 , 2 ,... , there exists some ( ) We construct a subsequence as follows. For = 1 , 2 ,... , let be the first term in which belongs to 1 / ( ). Then, ( ) is a subsequence of ( ) which converges to . We conclude that every sequence has a convergent subsequence. 1.119 Assume ( ) is a bounded sequence in . Without loss of generality, we can assume that { } [0 , 1]. Divide = [0 , 1] into two sub-intervals [0 , 1 / 2] and [1 / 2 , 1]. At least one of the sub-intervals must contain an infinite number of terms of the sequence. Call this interval 1 . Continuing this process of subdivision, we obtain a nested sequence of intervals 1 2 ......
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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.

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