Caratheodory's convex hull theorem

Suppose $K$ is a compact convex set in $\mathbb {R}^{n}$. Prove that every $x \in K$ is a convex combination of at most $n+1$ extreme points of $K$. Suggestion: Use induction on $n$. Draw a line from some extreme point of $K$ through $x$ to where it leaves $K$. Use Exercise 1.