Set of Rationals Greater than Root 2 has no Smallest Element

Theorem
Let $B$ be the set of all positive rational numbers $p$ such that $p^2 > 2$.

Then $B$ has no smallest element.

Proof
$p \in B$ is the smallest element of $B$.

Then by definition of $B$:
 * $p^2 > 2$

Let $q = p - \dfrac {p_2 - 2} {2 p}$.

Then $q > p$, and:

Hence:
 * $0 < q < p$

and so:

That means $q \in B$.

This contradicts our assertion that $p$ is the smallest element of $B$.

It follows that $B$ can have no smallest element.