Solution of Pell's Equation is a Convergent

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 $\sqrt n$.

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

Then we have:
 * $\paren {a - b \sqrt n} \paren {a + b \sqrt n} = 1$.

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

and so $a > b \sqrt n$.

Therefore:

The result follows from Condition for Rational to be Convergent.