In Distribution of class numbers in continued fraction families of real quadratic fields
Lemma 8 says
let $\omega_d=[u_0,\overline{u_1,u_2,\dots ,u_{s-1},u_s}]$ (note: only re-specified in the published version) $$\prod\limits_{i=0}^ju_i<\alpha_j<\prod\limits_{i=0}^j(u_i+1)$$
"The proof easily follows by induction from the recurrence $\alpha_{i+1}=u_{i+1}\alpha_i+\alpha_{i-1}$"
Note: $\omega_d=\sqrt d$ (page 6), $\alpha_{-1}=1$ and $\alpha_i=p_i+q_i\sqrt d$ with $\frac{p_i}{q_i}=[u_0,u_1,...,u_i]$
Now if I take $\sqrt {71}=[8,\overline{2,2,1,7,1,2,2,16}]$, the RHS does not seem to hold for the first few $j$
e.g. with $j=1$ we have $16<33.8523<27$
Is there something I missed? Or is there a typo in the paper?
Even the "easy induction" first step does not seem right: $u_0<u_0+\sqrt d<u_0+1$
Thanks
EDIT: I suspect that this is only true for purely periodic continued fractions.
In this case $\omega_d=[\overline{2a_0,u_1,\dots ,u_{s-1}}]=\sqrt d+a_0$
$-\overline{\omega_d}=\sqrt d-a_0$ and $\alpha_i=p_i+q_i(\sqrt d-a_0)$
In this scenario, the bounds and the induction (starting with $2a_0<a_0+\sqrt d<2a_0+1$) seem to be right