Hurwitz's Theorem (Number Theory)

Theorem
Let $\xi$ be an irrational number.

Then there are infinitely many relatively prime integers $p, q \in \Z$ such that:


 * $\left|{\xi - \dfrac p q}\right| < \dfrac 1 {\sqrt 5 \, q^2}$

Lemma 2
There are an infinite number of convergents to $\xi$.

Taking these in sets of $3$ at a time, it can be seen from Lemma 2 that there are an infinite number of approximations that satisfy the given inequality.

From Lemma 1 it is seen that this inequality is the best possible.