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This one always breaks my brain, or at least my intuition. I understand that the reals infinitely outnumber the rationals, and the diagonal proof makes perfect sense to me. But the "between two reals there is always a rational" just beats my intuition like there's candy inside it.

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Between any two reals, there’s a whole copy of R.

The original comment assumes its conclusion: it reduces to this smaller copy — then just jumps to the conclusion. It never actually tells us how to measure the ratio.

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.

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.

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