A real number is a class (a, say) of rational numbers which satisfies the condition that it is the same as the class of those rationals each of which precedes at least one member of a.
Therefore, that interval contains a rational q x and all those rationals are distinct.
Thus, consider the class of rationals less than 2r; any member of this class precedes some other members of the class - thus 1/2 precedes 4/3, 3/2 and so on; also the class of predecessors of predecessors of 2 2.
Note that the class of rationals less than or equal tò 2r is not a real number.
Consider the serial arrangement of the rationals in their order of magnitude.
