Prove inequality with norms and matrices

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.

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Is $\lVert \cdot \rVert$ a sub-multiplicative norm, that is $\lVert AB\rVert\leq \lVert A\rVert\lVert B\rVert$? –  Davide Giraudo Apr 11 '12 at 18:28

From your assumption, we know $B$ is invertible, since $||A-B||<||A^{-1}||^{-1}$ implies $||I-A^{-1}B||<1$ implying $A^{-1}B$ is invertible, so is $B$.
Firstly, I shall show $\|B^{-1}\|\le\frac{\lVert A^{-1} \rVert}{1-\lVert I-A^{-1}B\rVert}$. It suffices to show $\|B^{-1}\|\le \| A^{-1} \| +\lVert A^{-1}-B^{-1}\rVert$, which is obvious.
Now $$\lVert A^{-1} - B^{-1}\rVert= \lVert A^{-1}(A-B) B^{-1}\rVert \leq\lVert B^{-1}\rVert\lVert I-A^{-1}B\rVert\le\lVert A^{-1}\rVert \frac{\lVert I-A^{-1}B\rVert}{1-\lVert I-A^{-1}B\rVert}.$$ Done.