I'm going through Apostol's Calculus Volume 1 and the first few chapters are more about real analysis than calculus. You can use all the properties of elementary algebra, inequalities and the least upper bound axiom(and fundamental properties of the supremum and infinum-should I elaborate?) I would like hints on how to prove that given two non equal reals there exists one and hence an infinite number of rational and irrational numbers between them. I've proven that there exist an infinite number of reals in between the two non equal arbitrary real but I want hints for proving the other statements. Hints instead of a complete solution will be appreciated.
For the infinitude of rationals part try to find two rationals between the numbers.
then continuously average to show that there are an infinitude of rationals between them.
Have you proved that there are uncountably infinitely many reals between the two numbers or just infinitely many reals? The former makes the irrational part trivial to prove.