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.