Inequality condition for induced norms of iterative matrices

Question

Assume that $x^{(k+1)} = Bx^{(k)} + c$ and $||B||<1$ for some induced matrix norm and $x^{(0)} = x_0$. I need to prove:

$||x^{(k)} - x^*|| \geq \frac {||B^{k}||}{(I-||B||)}||x^{(1)}-x_0||$

However, I'm getting stuck at proving the following:

$||(I-B)^{-1}|| \leq \frac{1}{1-||B||}$

Currently all I know is

$\frac {1}{||I-B||} \leq ||(I-B)^{-1}||$

But I don't know if this is of any help.

And, $\frac {1}{1-||B||}>1$ because $||B||<1$

Please let me know how to proceed from here. Any help would be appreciated

Answer 1

Proof of the inequality $\|(I-B)^{-1}\| \leq \frac 1 {1-\|B\|}$: we have $(I-B)^{-1}=I+B+B^{2}+...$ so $\|(I-B)^{-1}\| \leq 1+\|B\|+\|B||^{2}+...=\frac 1 {1-\|B\|}$. These steps are valid since $\|B\| <1$.