Let $C$ be a fixed $n\times n$ matrix of real numbers and $b \in \mathbb{R}^n$ a fixed vector. Define $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ by $y = Tx = Cx+b$. I need to show that, using the Euclidean metric $d(x,z) = \sqrt{\sum\limits_{j=1}^n (\xi_j - \zeta_j)^2}$, $T$ is a contraction mapping if $\sum\limits_{j=1}^n \sum\limits_{k=1}^n c_{jk}^2<1$.
I feel like this should be pretty simple, but I'm struggling with the double-summation. $d(Tx,Tz) = \left[\sum\limits_{j=1}^n \left(\sum\limits_{k=1}^n c_{jk} (\xi_k-\zeta_k)\right)^2 \right]^{1/2},$ but I can't get from here to there in a way that I'm happy with.