I am trying to show that the closed unit ball in $C_0(R)$ has no extreme points. This is what I got so far and I am stuck. Please help me.
Suppose that $f \in C_0(R)$ and is an extreme point of the closed unit ball. Let $g(s) = f(s) + |f(s)| + 1$ and $h(s) = f(s) + |f(s)j| - 1$. Then $\|g\|_\infty \le 1$ and $\|h\|_\infty \le 1$ since $\|f\|_\infty \le 1$.
Is that the right way to start this problem?