There are plenty of simple proofs out there that $\sqrt{2}$ is irrational. But does there exist a proof which is not a proof by contradiction? I.e. which is **not** of the form: 

Suppose $a/b=\sqrt{2}$ for integers $a,b$. 

[*deduce a contradiction here*] 

$\rightarrow\leftarrow$, QED

Is it impossible (or at least difficult) to find a direct proof because **ir**-rational is a negative definition, so "not-ness" is inherent to the question? I have a hard time even thinking how to begin a direct proof, or what it would look like. How about: 

>$\forall a,b\in\cal I \;\exists\; \epsilon$ such that $\mid a^2/b^2 - 2\mid > \epsilon$.