Boundedness in Probability

Definition

$O_P(1)\iff \forall ϵ>0,\exists B\in {\mathbb{R}}^{+},\exists N\in \mathbb{N},\forall n\ge N,P(|X_n|\ge B)\le ϵ$

Facts

Let $\{X_n\}$ be a Random Vector, and $X$ be a Random Variable, then If $X_n\to X$, then $\{X_n\}$ is bounded in probability