Let $\left\lVert\cdot\right\rVert:\mathbb R^{n\times n}\to\mathbb R$ be a submultiplicative matrix norm and $A\in\mathbb R^{n\times n}$ such that $\lVert A\rVert<1$. Prove that $A+I_n$ is invertible, where $I_n$ is the identity matrix in $\mathbb R^{n\times n}$.
I tried coming up with something like $$\lVert A+I_n\rVert=\lVert A(I_n+A^{-1})\rVert\leq\lVert A\rVert\cdot\lVert(I_n+A^{-1})\rVert<\lVert I_n+A^{-1}\rVert,$$ but that doesn't seem to get me anywhere. In the end, I think I should have some (in)equality with the determinant of $I_n+A^{-1}$ in it (and conclude that it is not $0$), but I don't know how to get there. How could I proceed?