Prove that if $A$ is invertible and $||A-B||<||A^{-1}||^{-1}$ then
$\lVert (I-A)^{-1}\rVert \leq \frac{\lVert I\rVert-(\lVert I\rVert-1)\lVert A\rVert}{1-\lVert A\rVert}.$
This is the second part of the problem. I finally figured out the first part, but I am having trouble starting this one. Does anyone know what inequality I should start with to get this? Or the procedure I should take? I think with a little help I should be able to figure this one out.