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Unformatted text preview: 1.225 is closed and bounded (Proposition 1.1). 1. is bounded, that is there exists some such that ∥ x ∥ < for every x ∈ . Let x ∈ conv . x is a convex combination of a finite number of points in , that is x = ∑ =1 x with ∈ , ≥ 0 and ∑ =1 = 1. By the triangle inequality ∥ x ∥ ≤ ∑ =1 ∥ x ∥ < Therefore conv is bounded. 2. Let x belong to conv . Then, there exists a sequence ( x ) in conv which con- verges to x . By Carath´ eodory’s theorem, each term x is a convex combination of at most + 1 points, that is x = +1 ∑ =1 x where x ∈ . For each , the sequence ( x ) lies in a compact set and hence contains a conver- gent subsequence. Similarly, the sequence of coeﬃcients ( ) ∈ [0 , 1] is bounded and contains a convergent subsequence (Bolzano-Weierstrass theorem, Exercise...
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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.
- Fall '10