I am reading through a book about numerical linear algebra. It is challenging but I understand the concepts after some research. However, there is one thing that I can't figure out and it's about the connection between norms and invertibility in the "Perturbation theory" section.
Here is the lemma, which is referenced again and used to prove other theories:
This lemma is proved after, and everything is clear. However, it is used to prove further point and apparently is altered without any explanation:
We are getting $||A^{-1} δA|| <= 1$, after which we are given that Lemma 2.1 implies $I + A^{-1} δA$ Is invertible, in which it's not clear why the sign change happened, since we are given $I-X$ in the lemma. I have been trying to look up the explanations or come up with one by myself but without luck. Clearly the latter equation has many things different from the first one: We have inverse of the matrix $A^{-1}$, δA (the change in the matrix), we have $<=$ instead of $<$, and they are multiplied, but still none of the things explain the sign change, why is it $I + A^{-1}δA$ instead of $I - A^{-1}δA$.
The equation $I + A^{-1}δA$ is given without any explanation, as if it should be clear right away, so this question might have very simple and concise answer, but it's still unclear and any help is appreciated