Dedekind's Form of Completeness | Real Analysis

Dedekind's Form of Completeness:

Definition:

If X and Y are two non-empty subsets of R such that
(i) X U Y = R
(ii) There is no common element between X and Y.
(iii) x < y { x€ X and y€ Y}

Summary:
If all the real numbers divided into two non-empty sets X and Y such that every element of X is less than every element of Y,Then there exist a unique real number says ' a ' such that every real number belongs to X is less than 'a' and every real number greater than 'a' is belongs to Y.

Clearly,the two classes X and Y are disjoint and the number 'a' itself belongs to either to X or Y. Then either X has GLB or Y has LUB(completeness).
This property of real numbers is called as Dedekind's Completeness Property.