= Solution
Multiplying the matrices inductively gives columns $(p_n,q_n)^T$ and $(p_{n-1},q_{n-1})^T$. Their determinant is $(-1)^{n-1}$, and writing $x=(p_nx_{n+1}+p_{n-1})/(q_nx_{n+1}+q_{n-1})$ gives
$$
|x-p_n/q_n|={1\over q_n(q_nx_{n+1}+q_{n-1})}\leq{1\over q_nq_{n+1}}.
$$
If $Ay^2+By+C=0$, then for rationals $p/q$ near $y$, factorization against the conjugate root gives $|p/q-y|\geq M/q^2$; rationals away from $y$ are handled by reducing $M$. Finally $q_{n+1}\geq a_{n+1}q_n$, so the upper bound is at most $1/(a_{n+1}q_n^2)$. Unbounded partial quotients contradict the fixed lower bound for a quadratic irrational.
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