Trying to prove that there are no p and q such that $|\sqrt5 - p/q| < 1/(7q^2)$. Like the title says, I'm having trouble proving that there are no integers p and q such that
$|\sqrt5 - p/q| < 1/(7q^2)$. I was given the hint that $|(q\sqrt5 - p)(q\sqrt5 + p)| \geq 1$, but I don't quite know how that helps...
Thanks!

An alternative, we have $|q \sqrt{5} - p|< \tfrac{1}{7q}$ so if $ (q \sqrt{5}-p)(q \sqrt{5}+p) \geq  1 $ we can divide both sides
\begin{eqnarray}
\\
|q \sqrt{5} + p| & \geq &  7q \\
| \sqrt{5} + \tfrac{p}{q}| &\geq & 7
\end{eqnarray}
This is impossible since $| \sqrt{5} + \tfrac{p}{q}| \leq \sqrt{5} + \tfrac{p}{q} \leq \sqrt{5}+1 < 4$.
