Macroeconomics Exam Review 43

# Macroeconomics Exam Review 43 - 2 Since ∑ =1 ∣ ∣ = 1...

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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 coeﬃcient 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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