Question

How to prove that between every two real numbers ,there exists a rational number.

Answer 1

Answer:

Let x,y∈R, x≠y. Without loss of generality, suppose x<y. Then there exists a positive z such that y−x=z.

By Archimedes' axiom, there exists a natural number n such that

n>1z

nz>1

ny−nx>1

So there exists an integer m such that

nx<m<ny

x<mn<y

i.e. m/n is a rational number between x and y.

Since x and y can be any real numbers, in particular they can be irrationals..