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Homework Statement
Prove that every convex polytope is convex and closed.
Homework 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
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