Let $\alpha: V \rightarrow V$ be an endomorphism, and let $r\_k = $ rank($\alpha^k$). Show that $r\_k-r\_{k+1} \geq r\_{k+1}-r\_{k+2}$.

I have already shown that Im($\alpha^{k+1}$) $\leq$ Im($\alpha^{k}$) and Ker($\alpha^{k+1}$) $\geq$ Ker($\alpha^{k}$) for all $k$, and if $\alpha$ is an automorphism then equality holds throughout; but I can't seem to show why the rank has to obey the above. Rank-nullity doesn't seem to be helping.