During my Topology class the following question was brought up:
Question: Suppose that $A$ and $B$ are ordered countable sets. If there are two order preserving injections $f:A \to B$ and $g:B \to A$, is there an order preserving bijection between $A$ and $B$?
I do know that one can construct a bijection using two injections, it is a classic problem. However, I couldn't adapt the proof for order preserving injections.
False or not, I was hoping to find a proof for the following preposition:
Proposition: Every densely totally ordered countable set without lower or upper bound is isomorphic to $\mathbb{Q}$.
It appears that your conjecture as stated is false. Indeed, let $A=(0,1)\cap \Bbb Q$ and $B=\Bbb N \cup A$, $f:A\to B$ be the identity mapping, $g:B\to A$ be defined by $$ g(b)=\begin{cases} \frac b2\ ;\ b\in A \\ 1-\frac 1{b+1}\ ;b\in \Bbb N \end{cases}$$ , then both $f$ and $g$ are order preserving injections. However, $A$ and $B$ have different order type so there cannot be any order isomorphism between them.
Nevertheless, your proposition is true and, as Mr. BrianO has commented, was proved by Cantor. The wikipedia link is a nice place to visit but if you want a book, then Raymond Wilder's Introduction to the Foundations of Mathematics devotes some pages talking about order isomorphism of rational, real etc.