Legendre's theorem on continued fractions

Statement

Let $\alpha$ be a real number, and let $p$ and $q$ be positive integers such that $$\left| \alpha - \frac{p}{q} \right| < \frac{1}{2 q^2}.$$ Then $p / q$ is a convergent of the continued fraction expansion of $\alpha$.

Remarks

So the very good rational approximations of $\alpha$ are all convergents. The constant $\frac{1}{2}$ cannot be replaced by $1$: not every fraction with $\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}$ is a convergent. A use: it shows that the solutions of Pell's equation come from the convergents of $\sqrt{d}$.

History

Adrien-Marie Legendre proved the theorem in his Essai sur la théorie des nombres of 1798.

Explanations

L1 human

If a fraction is extremely close to $\alpha$, then the continued fraction algorithm finds it. So to look for the best approximations of a number, you only need to compute its convergents.