If $A, \phi \in \mathsf{GL}(V)$ where $V$ is a normed vector space and we are using the operator norm on $\mathsf{GL}(V)$, I'm trying to show that
$ \frac{ \|\phi^{-1}\|^{2} \| A - \phi \| }{1 - \| \phi^{-1}(A - \phi) \|} < \| \phi^{-1} \|^2 \|A - \phi\| $
whenever
$ \|A - \phi \| < \frac{1}{ \|\phi^{-1}\|} $
I have tried various approaches to this but none of them seem to work out. About the only thing I've been able to conclude is that the denominator is nonnegative but this doesn't seem to help with the estimation. I'm sure there's some algebra trick I could employ to see this but, if so, I cannot see it and would appreciate any constructive pointers on how to proceed.
If it helps, the context of this is a proof to show that the function that carries an operator to its inverse is continuous.