Solution
= Solution
With $p_{-1}=1,p_0=a_0$, $q_{-1}=0,q_0=1$ and $p_n=a_np_{n-1}+p_{n-2}$, $q_n=a_nq_{n-1}+q_{n-2}$, induction gives $p_nq_{n-1}-p_{n-1}q_n=(-1)^{n+1}$. <Matrix> multiplication gives $[a_0,\ldots,a_n,\beta]=(p_n\beta+p_{n-1})/(q_n\beta+q_{n-1})$; monotonicity in $\beta>0$ places it strictly between the adjacent convergents.