Let T be a linear operator of a n-dimensional vector space V over a field F. Suppose that T is nilpotent. Show that T ^ n = 0.

Let T be a linear operator of a n-dimensional vector space V over a field F. Suppose that T is nilpotent. Show that T ^ n = 0.

I have seen people prove this with an argument involving minimal polynomials. Can anyone prove this without such an argument?