A question on continued fraction

Question

Let $a$ be a positive irrational number. Let $p_k/q_k, p_{k+1}/q_{k+1}$ be two consecutive convergents of its simple continued fraction, where $k\ge 1$.

Is it possible that both $$|a-(p_k/q_k)|<1/(2q_k^2)$$ and $$|a-(p_{k+1}/q_{k+1})|<1/(2q_{k+1}^2)$$ are true?

I can only prove that at least one of these inequalities is true.

Answer 1

At least it can happen that not both are true. Example: 333/106 is a convergent to $\pi$, but $(\pi-(333/106)) \cdot 2 \cdot 106^2 \approx 1.87 > 1$.

Answer 2

It is known that $$|a-(p_k/q_k)|\le1/(q_kq_{k+1})\le1/(a_{k+1}q_k^2)$$ (see, e.g., Hardy and Wright) so if all the partial quotients exceed 2 then all the convergents satisfy your inequalitites.