Inverse of a sequence of random variables in Big O

Question

In a proof, the authors use that for a sequence of a random variable $X_n$ with $X_n=O_P(a_n)$, $a_n\rightarrow0$ and $a_n>0$, holds $X_n^{-1}=O_P(a_n^{-1}).$ I am not quite sure how to do that. My approach was to use the definition of $O_P$ $$\lim_{R\rightarrow\infty}\limsup_{n\rightarrow\infty}\mathbb{P}\left(\frac{\lvert X_n\rvert}{a_n}>R\right)=0.$$
Thank you for the help.