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Macroeconomics Exam Review 147

Macroeconomics Exam Review 147 - Solutions for Foundations...

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3.138 1. Since 𝑆 is open, there exists a ball 𝐵 𝑟 ( x 1 ) 𝑆 . Let 𝑡 = 1 + 𝑟 2 . Then x 0 + 𝑡 ( x 1 x 0 ) 𝐵 𝑟 ( 𝑥 1 ) 𝑆 . 2. Let 𝑠 = 𝑡 1 𝑡 𝑟 . The open ball 𝐵 𝑠 ( x 1 ) of radius 𝑠 centered on x 1 is contained in 𝑇 . Therefore 𝑇 is a neighborhood of x 1 . 3. Since 𝑓 is convex, for every y 𝑇 𝑓 ( y ) (1 𝛼 ) 𝑓 ( x ) + 𝛼𝑓 ( z ) (1 𝛼 ) 𝑀 + 𝛼𝑓 ( z ) 𝑀 + 𝑓 ( z ) Therefore 𝑓 is bounded on 𝑇 . 3.139 The previous exercise showed that 𝑓 is locally bounded from above for every x 𝑆 . To show that it is also locally bounded from below, choose some x 0 𝑆 . There exists some 𝐵 ( x 0 and 𝑀 such that 𝑓 ( x ) 𝑀 for every x 𝐵 ( x 0 ) Choose some 𝑥 1 𝐵 ( x 0 ) and let x 2 = 2 x 0 x 1 . Then x 2 = 2 x 0 x 1 = x 0 ( x 1 x 0 ) 𝐵 ( x 0 ) and 𝑓 ( x 2 ) 𝑀 . Since 𝐹 is convex 𝑓 ( x ) 1 2 𝑓 ( x 1 ) + 1 2 𝑓 ( x 2 ) and therefore 𝑓 ( x 1 ) 2 𝑓 ( x ) 𝑓 ( x 2 ) Since 𝑓 ( x 2 ) 𝑀 , 𝑓 ( x 2 ) ≥ − 𝑀 and therefore
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