What is a contraction on a space $(X,d)$? I have been reading some proofs on the elementary theorems of differential equations. One such proof uses the concept of a "contraction". See the definition below.

Definition 4 Let $(X,d)$ be a space equipped with a distance function $d$. A function $\Phi:X\to X$ from $X$ to itself is a contraction if there is a number $k\in (0,1)$ such that for any pair of points $x,y\in X$ we have
  $$d(\Phi(x),\Phi(y))\leq kd(x,y).$$
  It is important that the constant $k$ is strictly less than one.

Source. (Taken from this pdf.)
To put it into my own words, it seems that a contraction is simply a modified distance function that always yields a value less than the normal distance function for the same set of points. Am I understanding it correctly?
Would someone be able to offer an example of a contraction?
Thank you!

No, a contraction is not a metric(aka distance function). What a contraction does is bring every pair of points closer together(in the implicit metric). For instance, $f(x)=x/2$ is a contraction in the euclidean metric of $\mathbb{R}$.
