I'm given that a $F$ is square matrix and satisfies $\lVert F \rVert < 1$ for a norm that is submultiplicative.
Given this I'm asked to show that
$\lVert (I-F)^{-1} \rVert \leq \frac{1}{1-\lVert F \rVert}$
And the hint is to use the Neumann expansion which says that given conditions that we have that $(I-F)^{-1}=\sum_{i=0}^\infty F^i$
If I knew $\lVert (I-F)^{-1} \rVert \leq 1$ then this proof would be trivial because I could just multiply the left side by $1 - \lVert F \rVert$ and divide it over and I'd be done but I don't believe I know this is the case.