proof of properties of extreme subsets of a closed convex set

For the first claim, it is obvious that ${\bigcap }_{i\in I}{A}_i$ is closed convex subset (http://planetmath.org/ConvexSet) of $K$. Let $z\in A$ and , $x,y\in K$ such as $z=tx+(1-t)y$. Then $z\in A_i$, for all $i\in I$ so we have that $x,y\in A_i$ for all $i\in I$. Therefore $x,y\in {\bigcap }_{i\in I}{A}_i.$
For the second claim suppose $x,y\in K$, $t\in (0,1)$ and $z\in A$ such as $z=tx+(1-t)y$. From the hypothesis $A\subset B$ we have that $z\in B$ and since $B$ is an extreme subset of $K$, $x,y\in B$. Analogously from the hypothesis that $A$ is an extreme subset of $B$, we have that $x,y\in A$.

Title	proof of properties of extreme subsets of a closed convex set
Canonical name	ProofOfPropertiesOfExtremeSubsetsOfAClosedConvexSet
Date of creation	2013-03-22 15:25:12
Last modified on	2013-03-22 15:25:12
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	4
Author	georgiosl (7242)
Entry type	Proof
Classification	msc 52A99