What about this inequality involving two operators?

Question

Consider two operators $A$ and $B$ such that exists $M>0$ s. t. $\Vert A\Vert\leq M$. Moreover, let $k$ be a constant such that $\vert k\vert<\frac{1}{2}$. I want to show the following inequality $$\Vert \left(1+k A (B-i\eta)^{-1}\right)^{-1}\Vert \leq\left( 1 - M\vert k\vert \Vert (B-i\eta)^{-1}\Vert\right)^{-1},$$ where $\eta$ is a real number and $i$ is the imaginary unit. Could anyone please help me? Thank You in advance!