[thaumasiotes]

>> because there is an infinite quantity of irrational numbers between any two given rational numbers

> Indeed, you've grasped the core of it.

What? That's not the core of anything. It tells you that the irrationals are dense in the real number line. You know what other set is dense in the real line? The rationals.

[Tao3300]

I maintain that the rest still hinges on that observation. See other reply in thread.

[thaumasiotes]

How is that possible? We've made the same observation about the irrationals and the rationals. We want to make a followup observation that is true of the irrationals but not the rationals. Our first observation obviously can't be related.