I want to show that the function $f:\mathbb{R}\rightarrow \mathbb{R}$ with \begin{equation\*}f(x)=\frac{1}{2}\sin x-1\end{equation\*} is a contraction. 

A function $g : \Omega \rightarrow \Omega$ is a contraction, if it is Lipschitz-continuous with the Lipschitz-constant $L < 1$ in a vector norm. 

So we have to show that $$\|f(x)-f(y)\|\leq L\|x-y\|$$ Which norm do we have to use? The maximum norm?