I noticed this theorem mentioned in a book and I couldn't seem to find what it is called and/or see a proof of it. The closest I could find is the Drichlet Approximation Theorem.
The theorem in question is:
If $\vert{x - r/s}\vert < 1/2s^2$ for integers $r, s$ then ${r \over s} = {p_i \over q_i}$ for some $i$, where $p_i \over q_i$ is a finite termination at step $i$ of the continued fraction of $x$. In other words
$$ {p_i \over q_i} = a_0 + \frac{1}{a_1 + \frac{1}{... + \frac{1}{a_i}}} $$
Could somebody point me to a proof of this theorem and or tell me how I could start proving it?
A paper stated this as a "classic approximation relation" if that helps.
Theorem 19 of Khinchin's book, Continued Fractions.