Contraction of a Metric Space

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(X,d)$ be a metric space and let $T:X\to X$ be a map.

(i) For a real number $\lambda$ with $0\le\lambda<1$, $T$ is a contraction with constant $\lambda$ if $d(T(x),T(y))\le\lambda\,d(x,y)$ for all $x,y\in X$; and $T$ is a contraction if it is a contraction with constant $\lambda$ for some real number $\lambda$ with $0\le\lambda<1$.