Consider the sequence $\{\sqrt{n}|a_n-a|\}_n$ where $a_n, a \in \mathbb{R}$. Assume $\{\sqrt{n}|a_n-a|\}_n$ is bounded away from $0$ and $\infty$. Is this equivalent to or sufficient or necessary for
(1) $|a_n-a|=\frac{h}{\sqrt{n}}$ for some $0<h<\infty$?
(2) $|a_n-a|\in O(\frac{1}{\sqrt{n}})$?
Your sequences is bounded away from zero if and only if \begin{equation} \exists\, C_1> 0 \: \forall n \in \mathbb{N} \, : \: C_1 \leq \sqrt{n} |a-a_n|. \end{equation} This condition is necessary, but not sufficient for condition (1).
Your sequences is bounded from above (i.e. away from infinity) if and only if \begin{equation} \exists\, C_2 \ge 0 \, \forall n \in \mathbb{N} \: : \: \sqrt{n} |a-a_n| \leq C_2. \end{equation} This condition is equivalent to condition (2), which is really the statement \begin{equation} \exists\, C \ge 0 \, \forall n \in \mathbb{N} \: : \: |a-a_n| \leq \frac{C}{\sqrt{n}}. \end{equation}