Dedekind cut
In the Dedekind cut approach, each real number "x" is defined as the "infinite set of all rational numbers less than x". In particular, the real number 1 is the set of all rational numbers that are less than 1. Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers "r" such that "r" &lt; 0, or "r" &lt; 0.9, or "r" &lt; 0.99, or "r" is less than some other number of the form