Can someone give me some clue on how to show that rationals are well ordered? Thank you in advance.

Question

I wanted to show that the set of rationals are well ordered. A small hint would be really appreciated.

Answer 1

The rational numbers are not well-ordered. A set is well-ordered by a partial order $\prec$ if every non-empty set has a minimum element. The rational numbers with their natural order are not well-ordered, since the set itself has no minimum. There is no smallest rational number.

But the rational number can be well-ordered. Simply by showing that there is a bijection $f\colon\Bbb Q\to\Bbb N$, and then defining $q\prec p\iff f(q)<f(p)$. You can show that the order $\prec$ is indeed a well-ordering of $\Bbb Q$, but it is incompatible with the natural order of the rational numbers.