Prove that if $A$ is invertible and $||A-B||<||A^{-1}||^{-1}$ then
$\lVert A^{-1} - B^{-1}\rVert \leq \lVert A^{-1}\rVert \frac{\lVert I-A^{-1}B\rVert}{1-\lVert I-A^{-1}B\rVert}.$
I also need to prove
$\lVert (I-A)^{-1}\rVert \leq \frac{\lVert I\rVert-(\lVert I\rVert-1)\lVert A\rVert}{1-\lVert A\rVert}.$
I made several different attempts on starting to prove this problem without any success. I am not sure where the correct place to begin is. I think if I got a little direction on how to prove the first one I could get the second inequality.