Macroeconomics Exam Review 52

# Macroeconomics Exam Review 52 - iv Therefore x = â‰ z is an...

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Unformatted text preview: iv. Therefore ( x ) = â‰» ( z ) is an open neighborhood of x such that x â‰» z for every x âˆˆ ( x ) Similarly, ( z ) = â‰º ( x ) is an open neighborhood of z such that z â‰º x for every z âˆˆ ( z ). Consequently x â‰» z for every x âˆˆ ( x ) and z âˆˆ ( z ) 3. â‰¿ ( y ) = ( â‰º ( y ) ) (Exercise 1.56). Therefore, â‰¿ ( y ) is closed if and only if â‰º ( y ) is open (Exercise 1.80). Similarly, â‰¾ ( y ) is closed if and only if â‰» ( y ) is open. 1.237 1. Let = { ( x , y ) âˆˆ Ã— : x â‰¿ y } . Let (( x , y )) be a sequence in which converges to ( x , y ). Since ( x , y ) âˆˆ , x â‰¿ y for every . By assumption, x â‰¿ y . Therefore, ( x , y ) âˆˆ which establishes that is closed (Exercise 1.106) Conversely, assume that is closed and let (( x , y )) be a sequence converging to ( x , y ) with x â‰¿ y for every . Then (( x , y )) âˆˆ which implies that...
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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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