Suppose that then let . Now show that .
Let be the set of all positive rationals such that and let consist of all positive rationals such that . We want to show that there is no maximum element in and no minimum element in . So:
. Then .
First of all, how do we even know to do this? Just by intuition? Why did we set ? Could we have set it to something else? So we basically have shown that for every in , and for every in . By the way, this is from Rudin's "Principles of Mathematical Analysis."