How to show that if $\ker f^k =\ker f^{k+1}$ then $\ker f^{k+1} =\ker f^{k+2}$ where f is a linear map

Question

How to show that if $\ker f^k =\ker f^{k+1}$ then $\ker f^{k+1} =\ker f^{k+2}$ where f is a linear map ?

I have already shown that $\ker f^{k+1} \subset \ker f^{k+2}$

Answer 1

To show that $\ker f^{k+2}\subset \ker f^{k+1}$, let $x\in \ker f^{k+2}$. Then $f^{k+2}(x)=0$, or equivalently, $f^{k+1}(f(x))=0$. This implies that $f(x)\in\ker f^{k+1}=\ker f^k$ by assumption. Since $f(x)\in \ker f^k$, we have $0=f^k(f(x))=f^{k+1}(x)$, which gives $x\in\ker f^{k+1}$.