Macroeconomics Exam Review 43

Macroeconomics Exam Review 43 - 2. Since =1 = 1, 1 for...

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Unformatted text preview: 2. Since =1 = 1, 1 for every . Consequently, for every coordinate , the sequence ( ) is bounded. By the Bolzano-Weierstrass theorem (Exercise 1.119), the sequence ( 1 ) has a convergent subsequence with 1 1 . Let x , 1 denote the corresponding subsequence of x . Similarly, , 1 2 has a subsequence converging to 2 . Let ( x , 2 ) denote the corre- sponding subsequence of ( x ). Proceeding coordinate by coordinate, we obtain a subsequence ( x , ) where each term is x , = , x 1 + , x 2 + + , x and each coecient converges , . Let x = 1 x 1 + 2 x 2 + + 2 x Then x , x (Exercise 1.202). 3. Since =1 = 1 for every , =1 = 1. Consequently, at least one= 1....
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