I'd like to know if there are any relations between this $$||A^{-1}-B^{-1}||$$ and this $$||A-B||?$$ My supervisor is quite confident there must be one under certain conditions, but so far I've found nothing. I only managed to rewrite the expression in the following form $$tr(X^{-1}Y),$$ where Y is a positive semidefinite matrix. There are some inequalities for that case, but they ask quite a lot, can not afford :-)
p.s.: for clarification $A$ actually converges to $B$, that is $A=(A_n)_{n\ge1}:\lim A_n = B$. $||A||$ is a generic matrix norm, Frobenius norm e.g.
Observe that $$A^{-1}(B-A)B^{-1}=A^{-1}-B^{-1}.$$ Hence, if your norm satisfies that $$\lvert\lvert XY\rvert \rvert\le \lvert\lvert X\rvert \rvert\cdot \lvert\lvert Y\rvert \rvert$$ (which is the case for some natural norms but not all), then we have $$\lvert\lvert A^{-1}-B^{-1}\rvert \rvert\le \lvert\lvert B-A\rvert \rvert \cdot \lvert\lvert A^{-1}\rvert \rvert\cdot \lvert\lvert B^{-1}\rvert \rvert. $$
Hence, if $(A_n)_{n\in\mathbb{N}}$ is a sequence of invertible matrices that converges towards the invertible matrix $B$, then $({A_n}^{-1})_{n\in\mathbb{N}}$ converges towards $B^{-1}$.