I want to show that if $A=I+B$ and $||B||<1$ then $||A||||A^{-1}|| \leq \frac{1+||B||}{1-||B||}$, where $A\in\mathbb{C^{nxn}}$ and $||.||$ is a matrix norm
- We know that since $||B||<1$, $A$ is invertible
- I've shown that $1\leq||A||||A^{-1}|| \leq \frac{\big(||I||+||B||\big)||I||}{1-||B||}$
- and that $1\leq \frac{1+||B||}{1-||B||}\leq\frac{||I||+||B||}{1-||B||}\leq\frac{\big(||I||+||B||\big)||I||}{1-||B||}$
- and that $\rho(A)\leq\rho(I)+\rho(B)=1+\rho(B)<2$
But these don't seem very helpful.
Any hints?