Convergents converge to the number

Statement

Let $x$ be an irrational real number, and let $p_n / q_n$ be the convergents of its continued fraction expansion. Then $$\left| x - \frac{p_n}{q_n} \right| < \frac{1}{q_n q_{n+1}} \le \frac{1}{q_n^2}$$ for each $n \ge 0$, and $p_n / q_n \to x$ as $n \to \infty$.

Remarks

The convergents with even $n$ increase toward $x$, and the convergents with odd $n$ decrease toward $x$. For a rational $x$, the last convergent is equal to $x$. Conversely, each infinite simple continued fraction converges to an irrational number, and this number has that continued fraction as its expansion. For example, $\frac{355}{113}$ is a convergent of $\pi$, and $\left| \pi - \frac{355}{113} \right| < 3 \cdot 10^{-7}$.

Explanations

L1 human

The convergents of $x$ get closer and closer to $x$, and the error is smaller than one over the square of the denominator. This makes them very good rational approximations of $x$.