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Convex Polytope closedness

  1. Sep 22, 2011 #1

    zcd

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    1. The problem statement, all variables and given/known data
    Prove that every convex polytope is convex and closed.

    2. Relevant equations
    [tex] C=\{ \sum_{j=1}^n x_j a^j | x_j \geq 0, \sum_{j=1}^n x_j = 1\}[/tex] is a convex polytope

    3. The attempt at a solution
    I've already proven the convexity portion. To prove C is closed, I let [itex]\{ b^N \}_{N=1}^\infty \subseteq C[/itex] and assumed [itex]\lim_{N\to\infty} b^N = b[/itex].
    [itex]b=\sum_{j=1}^n x_j a^j[/itex], so I have to show [itex]\lim_{N\to\infty} x^N = x[/itex].

    I started with [itex]x_j \geq 0, \sum_{j=1}^n x_j = 1\ [/itex] means [itex] |x^N| \leq 1[/itex] and the sequence [itex]\{ x^N \}_{N=1}^\infty[/itex] is a bounded sequence. From here, I can use the Bolzano-Weierstrass theorem to show that there exists a subsequence that converges. From here, I'm unsure of what to do because the subsequence converges to some value which may or may not be the right value
     
  2. jcsd
  3. Sep 23, 2011 #2

    Office_Shredder

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    You know that the whole sequence converges to b, so every subsequence converges to b as well.
     
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