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.
Note that the class of rationals less than or equal tò 2r is not a real number.
Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of real numbers.
Consider the serial arrangement of the rationals in their order of magnitude.
Therefore, that interval contains a rational q x and all those rationals are distinct.