This theorem states that two unbounded countable dense linear orders are isomorphic.
[1] Suppose that Fix enumerations (without repetition) of the underlying sets: Now we construct a one-to-one correspondence between A and B that is strictly increasing.
Otherwise, we choose a suitable large or small element of B using the fact that B has neither a maximum nor a minimum.
According to Hodges (1993): While the theorem on countable densely ordered sets is due to Cantor (1895), the back-and-forth method with which it is now proved was developed by Edward Vermilye Huntington (1904) and Felix Hausdorff (1914).
Later it was applied in other situations, most notably by Roland Fraïssé in model theory.