I'm trying to think of a way to start this "proof":

Source:Let $\displaystyle S$ be an $\displaystyle n\times n$ strictly upper triangular matrix. Show that $\displaystyle \left(I-S\right)^{-1}=I+S^2+S^3+\dots+S^{n-1}$, 2nd Ed. by Schneider and BarkerMatrices and Linear Algebra

I don't want a complete proof. All I would appreciate would be an explanation on how I can start this. I will be able to take it from there.

Thank you in advance!!