A random variable $X_n$ satisfies $X_n=O_p(n^{-1/2})$. In my study, I defined an event $A_n$ such that $|X_n| \leq C_0 n^{-1/2}$ occurs for some positive constant $C_0$. I want to use the evnet $A_n$, so I came up with the following equivalent definition of $X_n=O_p(n^{-1/2})$.
$$\begin{aligned}X_n&=O_p(n^{-1/2}) \\\iff \lim_{C \rightarrow \infty} \limsup_{n \rightarrow \infty} P(|X_n| > C)&=0\\\iff \lim_{C \rightarrow \infty} \liminf_{n \rightarrow \infty} P(|X_n| \leq C)&=1\\\iff \liminf_{n \rightarrow \infty} P(|X_n| \leq C_0 a_n n^{-1/2})=1 \text{ for some constant $C_0 >0$ and $a_n \underset{n \to \infty}{\rightarrow}\infty$. }\end{aligned}$$ I wonder whether the very last equivalent statement holds.
