Let be a field and let be a finite dimensional vector space over . Let be a linear transformation. Prove that there exists a positive natural number such that , where denotes the th iterate of .
Note that if , then . Similarly, if , then . This gives us two chains of subspaces in , one ascending and one descending: and .
Note that because is finite dimensional, these chains must stabilize. Let and be minimal such that and , respectively, and let . We claim that .
Let . Now for some , and so . In particular, . So . Thus , as desired.