Let $U \subset \mathbb{R}^n$ be a nonempty convex [open set](/page/Open%20Set), and let $f \in C^1(U;\mathbb{R}^n)$ be a map such that $f(U) \subset U$. For each $x \in U$, let $\mathrm{d}f_x:\mathbb{R}^n\to\mathbb{R}^n$ denote the total derivative of $f$ at $x$. Suppose that there exists a constant $c \in [0,1)$ such that, for every $x \in U$,

\begin{align*} \|\mathrm{d}f_x\|_{\mathrm{op}} \le c. \end{align*}

Then $f:U \to U$ is a contraction with respect to the Euclidean metric; explicitly, for all $x,y \in U$,

\begin{align*} |f(x)-f(y)| \le c |x-y|. \end{align*}