Solution of Pell's Equation is a Convergent

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Theorem

Let $x = a, y = b$ be a positive solution to Pell's Equation $x^2 - n y^2 = 1$.

Then $\dfrac a b$ is a convergent of $\sqrt n$.

Let $a^2 - n b^2 = 1$.

Then we have:

$\left({a - b \sqrt n}\right) \left({a + b \sqrt n}\right) = 1$.

So:

$a - b \sqrt n = \dfrac 1 {a + b \sqrt n} > 0$

and so $a > b \sqrt n$.

Therefore:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left\vert{\sqrt n - \frac a b}\right\vert$	$=$	$\displaystyle \frac {a - b \sqrt n} b$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac 1 {b \left({a + b \sqrt n}\right)}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$<$	$\displaystyle \frac 1 {b \left({b \sqrt n + b \sqrt n}\right)}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac 1 {2 b^2 \sqrt n}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$<$	$\displaystyle \frac 1 {2 b^2}$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left\vert{\sqrt n - \frac a b}\right\vert\)	\(=\)	\(\displaystyle \frac {a - b \sqrt n} b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 {b \left({a + b \sqrt n}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(<\)	\(\displaystyle \frac 1 {b \left({b \sqrt n + b \sqrt n}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 {2 b^2 \sqrt n}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(<\)	\(\displaystyle \frac 1 {2 b^2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)