Given that $\lim_{n\to\infty}p_n/q_n=\sqrt{2}$, how to calculate $\lim_{n\to\infty}q_n|p_n-\sqrt{2}q_n|$?
Here, $p_n$ and $q_n$ are defined by the continued fraction of $\sqrt{2}$, and $p_n=a_np_{n-1}+p_{n-2}$ and $q_n=a_nq_{n-1}+q_{n-2}$, where $a_n$ is the $n$-th number in the continued fraction representation $\sqrt{2}=[1,2,2,\cdots]$.
I did a little bit of calculation for the first few terms and found out that the result should be $\sqrt{2}/4$, but I cannot figure out how to prove this result. It seems that ordinary single variable or multivariable approaches do not work in this case. Thanks!
Here it's easy to solve the recurrences in closed form: \begin{align} p_n&=\frac{(1+\sqrt2)^{n+1}+(1-\sqrt2)^{n+1}}{2} \\q_n&=\frac{(1+\sqrt2)^{n+1}-(1-\sqrt2)^{n+1}}{2\sqrt2} \end{align} (which makes $\lim\limits_{n\to\infty}q_n\big|p_n-q_n\sqrt2\big|=\frac1{2\sqrt2}$ obvious).