I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227.
A partial ordering $R$ is said to be dense iff whenever $xRz$, then $xRy$ and $yRz$ for some $y$. Assume $(A,R)$ is a linearly ordered structure with $A$ countable and $R$ dense. Show that $(A,R)$ is isomorphic to $(B,<_Q)$ for some subset $B$ of $\mathbb{Q}$.
I had an idea, but I'm not sure how to communicate it formally, so I'm hoping to get some help on this. First, since $A$ is countable, I can list it as a sequence, $A=\{a_0,a_1,a_2,\ldots\}$, even if this isn't the actual linear ordering dictated by $R$. The text suggests that I define some order isomorphism $f$ by defining $f(a_i)$ by recursion on $i$.
I began by letting $f(a_0)=q_0$ for some arbitrary rational. I then look at $a_1$, if $a_0Ra_1$, I would then let $q_0
Since $\mathbb{Q}$ is dense, I just want to choose some rational to preserve the order as I go along sequentially in $A$, and figure out the order as I go. Is this right? If so, what's the rigorous way to state this idea? I'm particularly concerned about how $f$ can respond to where the next element $a_{i+1}$ might be in relation to the previous $a_0,\ldots, a_i$, and how to decide which element to pick out. Thank you.