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 $\displaystyle a<b$ then

$\displaystyle 0< \frac{1}{b-a}$ by the archimedian principle we can pick an Integer N such that

$\displaystyle 0< \frac{1}{b-a}<N$ This implies that

$\displaystyle 1 < N(b-a)$ we can now pick another integer n such that

$\displaystyle n < N(a) < n+1$ adding 1 we get

$\displaystyle n+1 < N(a)+1$ usin g the fact that $\displaystyle 1 < N(b-a)$ we get

$\displaystyle n+1 <N(a) +1 < N(a) +N(b-a)=N(b)$ this implies that

$\displaystyle N(a) < n+1 < N(b) \implies a < \frac{n+1}{N}< b$