How do I show the following result for two commuting bounded idempotent linear operators on a normed space? I'm trying an old qualifier problem stating that for two commuting idempotent bounded linear operators $A$ and $B$ on a normed linear space, either $A=B$ or the operator norm of $A-B$ is at least one.
I'm at a complete loss as to how to approach this. Playing round with norms doesn't seem to get me anywhere and I don't see what other theorem to employ. 

Note $(A-B)^3 = A-B$ and so $||A-B|| \le ||A-B||^3 \implies A=B$ or $||A-B|| \ge 1$.
