A set partition of the Rational Numbers into two nonempty subsets and such that all members of are less than those of and such that has no greatest member. Real Numbers can be defined using either Dedekind cuts or Cauchy Sequences.
See also Cantor-Dedekind Axiom, Cauchy Sequence
Courant, R. and Robbins, H. ``Alternative Methods of Defining Irrational Numbers. Dedekind Cuts.'' §2.2.6 in
What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.
Oxford, England: Oxford University Press, pp. 71-72, 1996.