Convergents are Best Approximations/Corollary

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Corollary to Convergents are Best Approximations

Let $x$ be an irrational number.

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 $1 \le b \le q_n$


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


\(\ds \size {q_n x - p_n}\) \(=\) \(\ds q_n \size {x - \frac {p_n} {q_n} }\)
\(\ds \) \(>\) \(\ds q_n \size {x - \frac a b}\)
\(\ds \) \(\ge\) \(\ds b \size {x - \frac a b}\)
\(\ds \) \(=\) \(\ds \size {b x - a}\)

which contradicts the result of Convergents are Best Approximations.