Show that B is nilpotent if $[A,B]=B$

Question

Let A and B be any two complex matrices. Suppose that $AB-BA=B$. We want to prove that $B$ is nilpotent.

$B^2=BAB-B^2A$

$B^3=BAB^2-B^2AB$

$B^4=B^2AB^2-B^3AB$

$B^5=B^2AB^3-B^3AB^2$

Any hint would be appreciated. Thanks!