[CurtMonash]

There's a simple constructive proof using high-school level thinking. ... Two different reals have different decimal expansions. Go out far enough that they differ. Since this is about intuition, let's just assume the larger one is positive and irrational, and thus has an infinitely long expansion. Since the truncation has a finitely long decimal expansion, it's rational. And it's between the two original reals. Q.E.D. ... A full proof for all cases can be built similarly.

[gcanyon]

It doesn't break my brain that there's a rational between any two reals; it breaks my brain that this doesn't imply equivalent sizes between the reals and the rationals.