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?  

you could use the axiom of archimedes
