Macroeconomics Exam Review 75

Macroeconomics Exam Review 75 - 2.74 Let be defined as in...

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Unformatted text preview: 2.74 Let be defined as in Exercise 2.38. Let ( x ) be a sequence converging to x . Let = ( x ) and = ( x ). We need to show that → . ( ) has a convergent subsequence. Let ¯ = max and = min . Then ∈ [ , ¯ ]. Fix some > 0. Since x → x , there exists some such that ∥ x − x ∥ ∞ < for every ≥ . Consequently, for all ≥ , the terms of the sequence ( ) lie in the compact set [ − , ¯ + ]. Hence, ( ) has a convergent subsequence ( ). Every convergent subsequence ( ) converges to . Suppose not. That is, sup- pose there exists a convergent subsequence which converges to ′ . Without loss of generality, assume ′ > . Let ˆ = 1 2 ( + ′ ) and let z = 1 , z ′ = ′ 1 , ˆ z = ˆ 1 be the corresponding commodity bundles (see Exercise 2.38). Sincebe the corresponding commodity bundles (see Exercise 2....
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