= Stabilization of images of powers of a linear map
{title2=$\operatorname{im}h^m\supseteq\operatorname{im}h^{m+1}$}
= Stabilisation of images of powers of a linear map
{synonym}
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, $h^{m+k}v=h^m(h^kv)$ 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>.
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