Yo !

I've a problem about kernel and endomorphism. And i'm a little bit worried 'cause i'm blocked when i've just answered the first question. But let's see...

Ok, so we have $\displaystyle E$ a $\displaystyle \mathbb{K}$-vector space and $\displaystyle f$ an endomophism of $\displaystyle E$.

We have $\displaystyle f^0 = \text{Id}_E$ and $\displaystyle \forall k \geq 1, f^k = f \circ f^{k-1}$.

So, i've prooved that we always have, for all $\displaystyle k \geq 0$,

$\displaystyle \displaystyle \text{Im} f^{k+1} \subset \text{Im} f^k$ and

$\displaystyle \displaystyle \text{Ker} f^{k} \subset \text{Ker} f^{k+1}$

Now, the question is

Ok, now of course i've searche for a little while. But the fact is : i can't see how to start the proof. I think of several way : to prove it directly, maybe by induction, i've even think of trying it with contraposition !Show that, if $\displaystyle \text{Ker} f^{k} = \text{Ker} f^{k+1}$, then $\displaystyle \forall m\geq k, \text{Ker} f^{m} = \text{Ker} f^{k}$

Well, so i really don't want the answer ! Please !

But just thesmallest hint( such as, the way to prove it ) which can make me see how to keep going.

Thanks for reading me ! And i apologize if my english is poor

Hugal.