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
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