Theorem: If $A$ is a convergent matrix ($||A||<1$ for some natural norm) of size $n \times n$, then we have $||B_m-(I-A)^{-1}|| \leq \frac{||A||^{m+1}}{1-||A||}, m \in \mathbb{N}$, where $\text{{B}}^\infty_{m=0}$ is an infinite sequence of $n \times n$ matrices that converges to $(I-A)^{-1}$.
I've been looking at this for over an hour and I can't find a good place to start. I tried finding $||B_m-B_{m-1}||$ but got stuck. I'm just not sure how to develop $||A^{m+1}||$ at all.