Convergents are Best Approximations/Corollary

Corollary to Convergents are Best Approximations

Let $\sequence {p_n}_{n \mathop \ge 0}$ and $\sequence {q_n}_{n \mathop \ge 0}$ be the numerators and denominators of its continued fraction expansion.

Each convergent $\dfrac {p_n} {q_n}$ is a best rational approximation to $x$.

That is, for any rational number $\dfrac a b$ such that $1 \le b \le q_n$:

$\size {x - \dfrac {p_n} {q_n} } \le \size {x - \dfrac a b}$

The equality holds only if $a = p_n$ and $b = q_n$.

Assume otherwise:

$\exists \dfrac a b$ such that $1 \le b \le q_n$

and:

$\size {x - \dfrac {p_n} {q_n} } > \size {x - \dfrac a b}$

Then:

$\ds \size {q_n x - p_n}$

$=$

$\ds q_n \size {x - \frac {p_n} {q_n} }$

$\ds $

$>$

$\ds q_n \size {x - \frac a b}$

$\ds $

$\ge$

$\ds b \size {x - \frac a b}$

$\ds $

$=$

$\ds \size {b x - a}$

$\blacksquare$