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Macroeconomics Exam Review 162

Macroeconomics Exam Review 162 - Solutions for Foundations...

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𝐴 𝐵 Figure 3.2: 𝐴 and 𝐵 cannot be strongly separated. for every z 𝑆 (Exercise 3.182). For every x 𝐴, y 𝐵 , z = y x 𝑆 and 𝑓 ( z ) = 𝑓 ( y ) 𝑓 ( x ) 𝑐 > 0 or 𝑓 ( x ) + 𝑐 𝑓 ( y ) which implies that sup x 𝐴 𝑓 ( x ) + 𝑐 inf y 𝐵 𝑓 ( y ) and sup x 𝐴 𝑓 ( x ) < inf y 𝐵 𝑓 ( y ) 3.195 No. See Figure 3.2. 3.196 1. Assume that there exists a convex neighborhood 𝑈 0 such that ( 𝐴 + 𝑈 ) 𝐵 = Then ( 𝐴 + 𝑈 ) is convex and 𝐴 int ( 𝐴 + 𝑈 ) = and int ( 𝐴 + 𝑈 ) 𝐵 = . By Corollary 3.2.1, there exists continuous linear functional such that 𝑓 ( x + u ) 𝑓 ( y ) for every x 𝐴, u 𝑈, y 𝐵 Since 𝑓 ( 𝑈 ) is an open interval containing 0, there exists some u 0 with 𝑓 ( u 0 ) = 𝜖 >
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