Proving a unique solution with the norm of a matrix in a linear system

Question

Let $B\in\mathcal{M}_n$ with $\left \| B \right \|<1$ and $c$ a vector in $\mathbb{R}^n$. Prove that the linear system:

$u=Bu+c$

Has a unique solution $u$

I know how to prove a unique solution, but I don't know how to use the norm of the matrix to prove it.

Answer 1

Elaborating on Kavi Rama Murthy's comment: the eigenvalues of $I-B$ are of the form $1-\lambda$ where $\lambda$ is an eigenvalue of $B$. If $\|B\| < 1$, then $|\lambda| < 1$ for all eigenvalues of $B$. This implies $0$ is not an eigenvalue of $I-B$.