Fix enumerations (without repetition) of the underlying sets:
Now we construct a one-to-one correspondence between A and B that is strictly increasing. Initially no member of A is paired with any member of B.
It still has to be checked that the choice required in step (1) and (2) can actually be made in accordance to the requirements. Using step (1) as an example:
If there are already ap and aq in A corresponding to bp and bq in B respectively such that ap < ai < aq and bp < bq, we choose bj in between bp and bq using density. Otherwise, we choose a suitable large or small element of B using the fact that B has neither a maximum nor a minimum. Choices made in step (2) are dually possible. Finally, the construction ends after countably many steps because A and B are countably infinite. Note that we had to use all the prerequisites.
If we iterated only step (1), rather than going back and forth, then in some cases the resulting function from A to B would fail to be surjective. In the easy case of unbounded dense totally ordered sets it is possible to avoid step 2 by choosing the element bj more carefully (by choosing j as small as possible), but this does not work for more complicated examples such as atomless Boolean algebras where steps 1 and 2 are both needed.
According to Hodges (1993):
See also: Ehrenfeucht–Fraïssé game.
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