# Convergence in distributiom

I need to show that for arbitrary random variables $X_n$, there exist a sequence of positive constants $a_n$ such that $a_nX_n\overset{D}\rightarrow 0$.

Thus, I need to show that $\lim_{n\rightarrow \infty} P(a_nX_n\leq x)=\begin{cases} 0\text{ if } x<0\\1 \text{ if } x>0\end{cases}$ or at lest $\lim_{n\rightarrow \infty} P(a_nX_n>\epsilon)\rightarrow 0$ for all $\epsilon>0$.

I can show this for finite random variables by taking infimum over all $\epsilon>0$, but have no clue how to show it for general random variables. Any thoughts?

Since $\lim_{x\rightarrow+\infty}P\left(\left|X_{n}\right|\leq x\right)=1$ you can choose positive $a_{n}$ such that: $$P\left(\left|X_{n}\right|\leq\frac{a_{n}^{-1}}{n}\right)>1-\frac{1}{n}$$ or equivalently: $$P\left(a_{n}\left|X_{n}\right|\leq\frac{1}{n}\right)>1-\frac{1}{n}$$

Based on that for every fixed $x>0$ it can be shown that: $$\lim_{n\rightarrow\infty}P\left(a_n\left|X_{n}\right|\leq x\right)=1$$