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 $\mathbb{Q}$.