This is probably trivial, but a lot of time has passed since I studied these topics, so I'm a bit rusty.
So, let $\mathbf{A}$ be square matrix in $\mathbb{R}^{n\times n}$. We want to prove that there exists $c\in\mathbb{R}_{>0}$ such that the linear map $\mathbf{I}-c\mathbf{A}$ is a contraction. I need to prove that $\forall\ \mathbf{v},\mathbf{w}\in\mathbb{R}^{n}$, there exists $k\in[0,1[$ such that
$$||(\mathbf{I}-c\mathbf{A})(\mathbf{v}-\mathbf{w})||\leq k||\mathbf{v}-\mathbf{w}||$$
which reduces to proving that $\forall\ \mathbf{v}\in\mathbb{R}^{n}$,
$$||(\mathbf{I}-c\mathbf{A})\mathbf{v}||\leq k||\mathbf{v}||$$
Now, if I use an induced norm, I know that
$$||(\mathbf{I}-c\mathbf{A})\mathbf{v}||\leq ||\mathbf{I}-c\mathbf{A}||\cdot||\mathbf{v}||$$
so it all amounts to proving that $||\mathbf{I}-c\mathbf{A}||<1$. Now, from the triangle inequality and using the fact that $||\cdot||$ is an induced norm, I can get
$$1-c||\mathbf{A}||\leq||\mathbf{I}-c\mathbf{A}||\implies c||\mathbf{A}||\geq1-||\mathbf{I}-c\mathbf{A}||$$
but this doesn't seem to help me much. What am I missing?