Let $A$ be an abelian C*-algebra and $p$ be a projection in $A$. To show $p$ is an extreme point of $A^+_{\|.\|\leq 1}$ suppose there is $b,c\in (A^+)_{\|.\|\leq 1}$ such that $p= \frac{1}{2}(b+c)$ then $b,c \in (pA)_{\|.\|\leq 1}$, because $0\leq b(1-p)\leq p(1-p)=0$ and $0\leq c(1-p)\leq p(1-p)=0$.
I think $b\leq p$ implies $b(1-p)\leq p(1-p)$, but I can not show $b\leq p$ and $c\leq p$. Please give me a hint. Thanks.