Non-Constructive Proofs

Non-constructive proofs are a neat trick to show the existence of something without providing an example.

A common example for such a proof is for the theorem «there exist two irrational numbers $a$ and $b$ such that $a^b$ is rational.» It goes as follows:

Let $a = b = \sqrt{2}$

$a^b = \sqrt{2}^{\sqrt{2}}$ is either rational or irrational

  1. If it is rational then our statement is proved.
  2. If it is irrational, then we can pick $a = \sqrt{2}^{\sqrt{2}}$.
    $(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^2 = 2$, which is rational.