Let $\theta=[a_0;a_1,a_2,\dots]$ be an irrational [continued fraction](/page/Continued%20Fraction), and let $p_n/q_n$ denote its $n$th convergent. Then the even convergents form an increasing sequence, the odd convergents form a decreasing sequence, and every even convergent is less than every odd convergent. More explicitly, for all $r,s \geq 0$,

\begin{align*} \frac{p_{2r}}{q_{2r}} < \frac{p_{2r+2}}{q_{2r+2}}, \end{align*}

\begin{align*} \frac{p_{2r+1}}{q_{2r+1}} > \frac{p_{2r+3}}{q_{2r+3}}, \end{align*}

and

\begin{align*} \frac{p_{2r}}{q_{2r}} < \frac{p_{2s+1}}{q_{2s+1}}. \end{align*}