 Stabilization of images of powers of a linear map

For a linear endomorphism $h$ of an $n$-dimensional vector space, the images of a linear map formed by its powers decrease and stabilize by exponent $n$. Indeed, $h^{m+k}v=h^m(h^{k}v)$ proves inclusion. If equality occurs at one step, applying $h$ to both sides proves equality at the next step, and induction gives equality forever after. Before stabilization, every inclusion is strict and lowers dimension by at least one. Starting at dimension $n$ allows at most $n$ such decreases.

By the rank-nullity theorem, the increasing null spaces of the same powers stabilize at the same time. This gives a short proof of the finite-dimensional Fitting decomposition.