I have to prove that:
If the space V is finitely generated, all the generating systems of V are finite.
Therefore, I have tried doing the following: I know that if V is finitely generated, exists a finite generator system of V and V has got a finite basis. I also know that if V is not {0}, all the bases of V have the same dimension. But I don't really get a solid conclusion.
I also have an other idea, which I'm not sure it's ok: If V is finitely generated, we supposed that $\dim V=n$. As the dimension of the generator system has to be equal or bigger that $\dim V$, the proposition is false. (?)
You cannot prove it, since it is false. The whole space $V$ itself is a generating set of $V$ and, in general, it is infinite.