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 $x,y\in A$.