Kernel/Image of a repeated linear transformation

Let V be a vector space of finite dimension (say n). A:V->V is a linear transformation. prove there exists an integer m such that the intersection of im(A^m) and ker(A^m) is {0}.

since the image and kernel are subspace, {0} is always in both.

I know the general strategy is to use dim(im(A^m))+dim(ker(A^m))=n and that after successive applications of a linear transformation the kernel gets smaller. I just don't know how to prove that the kernel gets smaller. I tried to show that dim(ker(A^m))<dim(ker(A^(m-1))) without success.

Re: Kernel/Image of a repeated linear transformation

He's right, all you know is that $A^{k+n}(y)=0$ for all $n$. To fix, say $A^k(x)=y\Rightarrow A^{2k}(x)=A^k(y)=0$. Then you have $x\in\text{Ker}\phi^{2k}$, but $\text{Ker}\phi^{2k}=\text{Ker}\phi^k$ so $x\in\text{Ker}\phi^k$.