If there are already "a""p" and "a""q" in "A" corresponding to "b""p" and "b""q" in "B" respectively such that "a""p" < "a""i" < "a""q" and "b""p" < "b""q", we choose "b""j" in between "b""p" and "b""q" 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.