Let a and b are two rational numbers such that a<b .
`a,b in Q` , Q is set of rational numbers.
`a=p/q and b=r/s`
by def of rational numbers .
`=(1/2)((ps+rq)/(qs)) in Q`
`a<(a+b)/2<b` , this proves there exist at least one rational number between a and b.