If $A$ is a non-empty, compact and convex set in a finite dimension space, $B\subset A$ and $\text{cl(conv}(B))=A$. $\text{ext}(A)$ denotes the set of extreme points of $A$. Prove that $\text{ext}(A)\subset \text{cl}(B)$.
Suppose $a\in\text{ext}(A)$ is not in $\text{cl}(B)$. When $B$ is convex from Hyperplane separation theorem we derive $\text{cl(conv}(B))\neq A$. When $B$ is concave, there is a supporting hyperplane at $a$ and $A$ is in one side of the plane, then how to derive contradition?