pandaman64

Let's call (r_i)_{i>=0} the sequence of the ranks of (A^i)_{i>=0}.
For any i>=0, the image of A^{i+1} is a subspace of the image of A^{i}.
Our sequence of ranks (r_i) is non-increasing.

Since Im(A^{i+1}) ⊂ Im(A^i), if there exists i such that r_i = r_{i+1}, Im(A^{i+1}) = Im(A^i).
From this, it follows that Im(A^k) = Im(A^i) (k > i), or r_k = r_i.

In other words, (r_i) is a sequence of non-negative integers that 
- starts at n
- might be stricly decreasing for a while