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Macroeconomics Exam Review 161

Macroeconomics Exam Review 161 - Solutions for Foundations...

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3.189 Let x 𝐶 . Since 𝐶 is a cone, 𝜆 x 𝐶 for every 𝜆 0 and therefore 𝑓 ( 𝜆 x ) 𝑐 or 𝑓 ( x ) 𝑐/𝜆 for every 𝜆 0 Taking the limit as 𝜆 → ∞ implies that 𝑓 ( x ) 0 for every x 𝐶 3.190 First note that 0 𝑍 and therefore 𝑓 ( 0 ) = 0 𝑐 so that 𝑐 0. Suppose that there exists some z 𝑍 for which 𝑓 ( z ) = 𝜖 = 0. By linearity, this implies 𝑓 ( 2 𝑐 𝜖 z ) = 2 𝑐 𝜖 𝑓 ( z ) = 2 𝑐 > 𝑐 which contradicts the requirement 𝑓 ( z ) 𝑐 for every z 𝑍 3.191 By Corollary 3.2.1, there exists 𝑓 𝑋 such that 𝑓 ( z ) 𝑐 𝑓 ( x ) for every x 𝑆, z 𝑍 By Exercise 3.190 𝑓 ( z ) = 0 for every z 𝑍 and therefore 𝑓 ( x ) 0 for every x 𝑆 Therefore 𝑍 is contained in the hyperplane 𝐻 𝑓 (0) which separates 𝑆 from 𝑍 . 3.192 Combining Theorem 3.2 and Corollary 3.2.1, there exists a hyperplane
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