Let S be a compact subset of a finite-dimensional linear space X of dimension n.
1. Show that conv S is bounded.
2. For every x ˆˆ conv S, there exists a sequence (xk) in conv S that converges to x (exercise 1.105). By CaratheÂodory's theorem (exercise 1.175), each term xk is a convex combination of at most n + 1 points, that is,

Where xki ˆˆ S. Show that we can construct convergent subsequences aki †’ ai and xki †’ xi.
3. Define x =

Show that xk †’ x.
4. Show that x A conv S.
5. Show that conv S is closed.
6. Show that conv S is compact.

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