Proof that if A is invertible and $||A-B|| < ||A^{-1}||^{-1}$ then $||A^{-1}-B^{-1}||$ $\leq$ $||A^{-1}||\frac{||I-A^{-1}B||}{1-||I-A^{-1}B||}$
I have tried substuting in $||A^{-1}-B^{-1}|| = B^{-1}(B-A)A^{-1}$ and chaging $||I-A^{-1}B||$ to $||A^{-1}||||A-B||\leq I $from the given condition, but I cant get final inequality.
Note that $B$ is invertible from $A$ invertible and $\lVert A-B\rVert\leq\lVert A^{-1}\rVert^{-1}$. If $A=B$ there is nothing to prove, so assume $A\neq B$.
From $A^{-1}-B^{-1}=(I-B^{-1}A)A^{-1}$, you have $\lVert A^{-1}-B^{-1}\rVert\leq\lVert A^{-1}\rVert\cdot\lVert I-B^{-1}A\rVert$. Now we want: $$ \lVert I-B^{-1}A\rVert\leq\frac{\lVert I-A^{-1}B\rVert}{1-\lVert I-A^{-1}B\rVert} $$ or equivalently, $$ \lVert I-B^{-1}A\rVert\leq\lVert I-A^{-1}B\rVert+\lVert I-A^{-1}B\rVert\lVert I-B^{-1}A\rVert $$ which follows from $$ I-B^{-1}A=(I-A^{-1}B)(I-B^{-1}A)-(I-A^{-1}B). $$