Let $M$ be a $2 \times 2$ real matrix such that $$\parallel{}Mx\parallel{} \leq \frac{1}{4}\parallel{}x\parallel{}$$
for all $x \in \mathbb{R}^2$ where $\parallel{}\parallel{}$ is the euclidean norm on $\mathbb{R}^2$. I want to show that $$K(u):=||u||Mu+f \;,\;\;\;\text{ } ||f||\leq \frac{3}{4}$$ is a contraction on the closed unit disc $B_1(0)$. Can someone give me a hint since I have no idea how to get further than the last inequality: \begin{align} ||K(u)-K(v)|| &= \left|\left|M||u||u-M||v||v\right|\right|\\ &= \left|\left|M(||u||u-||v||v)\right|\right|\\ &\leq \frac{1}{4}||(||u||u-||v||v)|| \end{align}