For the quadratic surd:
$$ \zeta = \dfrac{P + \sqrt D}Q $$
the wikipedia article on periodic continued fractions mentions that Lagrange proves that the largest partial denominator of a regular continued fraction for $\zeta$ is less than $2\sqrt D$. I also note that for this continued fraction:
$$ \sqrt D = [a_0;\overline{a_1,a_2,\dots a_2,a_1,2a_0}] $$ and $$ 2a_0 = 2\lfloor\sqrt D\rfloor < 2\sqrt D $$
which would seem to be a clue to the bound, as $2a_0$ is close to but less than the bound. However, that pattern is only shown for $\sqrt D$, not for quadratic surds in general. So how was this bound derived and proven by Lagrange? I haven't found anything pertinent on Google or other math sites explaining this.
Edit
In light of the fact a large $P$ would seem to disconnect the size of $a_0$ from the size of $2D$, I'm going to assume that Wikipedia stated this poorly, and was only making a statement about $\sqrt D$. In that case, how is the $2\sqrt D$ bound proven for $\sqrt D$?