Question

Prove: for any two real numbers that are not equal, you can find a real number between them. (You need to prove that the real number you select is between the original two numbers, and your proof needs to be elementary, i.e. only rely on college-algebra level work with inequalities. You cannot use advanced concepts such as the Archimedean property.)

 

discrete math

Answer 1

If the two numbers are a and b, and a<b, then there is another real number between them, x=(a+b)/2=a/2+b/2.

x-a=(a/2+b/2)-a=b/2-a/2 and b-x=b-(a/2+b/2)=b/2-a/2=(b-a)/2>0. Therefore x is equidistant from a and b.

Since a<b, b-a>0. x-a>0 and b-x>0 so x>a and b>x making a<x<b.