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.

Proof that there is a rational between any two real 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