Thread: Proof involving rational numbers

a) Prove that between any two rational numbers there is another rational number; that is, if a, b are in Q and a < b, then there exists z in Q such that a < z < b.

b) Prove that between any two rational numbers there are infinitely many rational numbers.

Since then

by the archimedian principle we can pick an Integer N such that

This implies that

we can now pick another integer n such that

adding 1 we get

usin g the fact that we get

this implies that

You don't have to be that complicated! If a and b are rational numbers, then is a rational number between a and b.

Suppose there were only a finite number of rational numbers between a and b. Then there must be largest such number: c. But (b+c)/2 is a rational number between c and b and so between a and b and is larger than c: contradiction.