Prove that there is an infinite number of rationals between any two reals I just stumbled upon this question: Infinite number of rationals between any two reals..
As I' not sure about my idea of a proof, I do not want to post this as an answer there, but rather formulate as a question.
My idea is as follows:


*

*$\mathbb{Q} \subset \mathbb{R}$

*$\forall a,b \in \mathbb{R}$ with $a>b, \exists q_0 \in \mathbb{Q}$ s.t. $a > q_0 > b$ (which is proven e.g. on Proofwiki)

*As $\mathbb{Q} \subset \mathbb{R}, q_0 \in \mathbb{R}$

*For $a, q_0$, repeat step 2 to find $q_1 \in \mathbb{Q}$ s.t. $a > q_1 > q_0 > b$

*Repeat ad infinitum


Thus, there have to be infinitely many rationals between any two reals.
Can you argue like this, or is there anything wrong in my line of reasoning?  

I was also thinking along similar lines : Take a rational upper bound A for a, and a rational lower bound B for b so that $a<A<B<b$, then use the fact that the mean M of two rational numbers is itself rational, since $\mathbb{Q}$ forms a ring with addition (+) and multiplication ($\cdot$) , and then apply it with regards A & M and B & M to find the two new means $M_A$ and $M_B$, and so on and so forth, ad infinitum.
