Recently I found a lemma (without proof) which stated that the sum of a regular matrix $A \in \mathbb{R}^{n \times n}$ and another matrix $S$ of the same type is invertible if $\|A\|\|S\|<1$ and that in that case the following inequality holds. $ \|(A+S)^{-1}\| \leq \frac{\|A\|}{1-\|A^{-1}\|\|S\|}$
I would be more interested in understanding why under the mentioned condition the inverse exists. It kinda makes sense that by distorting an invertible matrix with another "small" matrix the result should still be invertible. However I did not find a proof or any insights as to why that is the case.
It all comes from considering the equation $$(A + S) \sum_{k=0}^\infty (-1)^k (A^{-1} S)^k A^{-1} = I$$ $\|A^{-1} S\| < 1$ is what you need to make the series converge.