TITLE: Is there a rational number between any two irrationals?
QUESTION [53 upvotes]: Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$.  Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$?  How would you construct such a rational number?
[I posted this only so that the useful answers at https://math.stackexchange.com/questions/414036/rationals-and-irrationals-on-the-real-number-line/414048#414048 could be merged here before that question was deleted.]

REPLY [11 votes]: You can apply an "argument from authority" here :). As we all know, Dedekind constructed a one-to-one correspondence between the real numbers and certain sets of rational numbers, which he called "cuts". We all accept that Dedekind's construction of the real numbers is mathematically flawless. By the way these cuts are constructed (see his 1872 paper), if there would be two irrational numbers without a rational number in between, then these two irrational numbers would correspond to the same Dedekind Cut. That would imply that there is no one-to-one correspondence between the real numbers and his cuts, hence Dedekind's construction of the reals would be incorrect (--> contradiction).