Hurwitz’ theorem

Abstract:

If $[{a_0},{a_1},{a_2}, \cdots ]$ is the continued fraction for a real number $x$, and ${p_n}/{q_n}$ the $n$th convergent, define ${\theta _n} = {q_n}|{p_n} - x{q_n}|$. Hurwitz’ Theorem asserts that ${\phi _n} = \min \{ {\theta _{n - 1}},{\theta _n},{\theta _{n + 1}}\} < {5^{ - 1/2}}$ whenever ${\phi _n}$ is defined. It is the object of this note to provide a simple proof of this fact.