I have to prove the following theorem using the induction on $n$:
Every compact subset $K$ in $\mathbb{R}^n$ has extreme points, and every point of $K$ can be written as a convex combination of $n+1$ extreme points.
I think I have to show two things.
(i) for $n=1$ the above theorem holds.
(ii) Assuming that the above theorem holds for $n$, I have to prove it for $n+1$.
However, I am stuck at (ii). How can I derive the theorem for $n+1$ from $n$? I think I have to use the projection map, but cannot proceed. Could anyone help me?