For $p,q\in \mathbb{Q} :p<q $ we define $(p,q) :=\{ x\in \mathbb Q : p<x<q \}$ and we want to define an order like this $$(p,q) < (t,l) \iff p<q$$ and I want to prove that there is an order perserving function from $\langle \mathbb{Q} ,<_{\mathbb Q} \rangle$ to $S = \{ (p,q): p,q\in \mathbb Q , p<q \}$ with the order we defined.
I understood that I have to define a family of functions that the union on them is the function that I want. But I'm having some troubles with the induction assumption, what conditions I have to assume to find such a function. I'll be really glad to hear how you knew that these are the right assumptions.
thank you very much