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Macroeconomics Exam Review 91

# Macroeconomics Exam Review 91 - c 2001 Michael Carter All...

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𝑓 𝑔 ( x ) 𝑆 for every x 𝐵 , we must have 𝑓 𝑔 ( x ) = x 𝑆 . Therefore, 𝑔 ( x ) = x which implies that 𝑓 ( x ) = x . That is, x is a fixed point of 𝑓 . Brouwer = no-retraction Exercise 2.132. 2.135 Let Λ 𝑘 , 𝑘 = 1 , 2 , . . . be a sequence of simplicial partitions of 𝑆 in which the maximum diameter of the subsimplices tend to zero as 𝑘 → ∞ . By Sperner’s lemma (Proposition 1.3), every partition Λ 𝑘 has a completely labeled subsimplex with vertices x 𝑘 0 , x 𝑘 1 , . . . , x 𝑘 𝑛 . By construction of an admissible labeling, each x 𝑘 𝑖 belongs to a face containing x 𝑖 , that is x 𝑘 𝑖 conv { x 𝑖 , . . . } and therefore x 𝑘 𝑖 𝐴 𝑖 , 𝑖 = 0 , 1 , . . . , 𝑛 Since 𝑆 is compact, each sequence x 𝑘 𝑖 has a convergent subsequence x 𝑘 𝑖 . Moreover, since the diameters of the subsimplices converge to zero, these subsequences must converge to the same point, say x . That is, lim 𝑘 →∞ x 𝑘 𝑖 = x , 𝑖 = 0 , 1 , . . . , 𝑛
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