liminf and limsup in probability 
Consider a sequence of random variable $(X_n)$. Prove the following inequality:
  $$
\mathbb P\left(\liminf\{X_n \leq x\}\right) \leq \liminf\mathbb P\left(\{X_n \leq x\}\right)\leq \limsup\mathbb P\left(\{X_n \leq x\}\right) \leq \mathbb P\left(\limsup\{X_n \leq x\}\right).
$$

I can see that the second $\leq$ is simply the result for real sequence. However, I do not see how to get the first and the last one. It seems that the two are similar. Could anyone help me, please? Thank you!

This has nothing to do with random variables. Just let $A_n = \{X_n \leq x\}$
$$
\mathbb P\left(\liminf A_n\right) \leq \liminf\mathbb P\left(A_n\right)\leq \limsup\mathbb P\left(A_n\right) \leq \mathbb P\left(\limsup A_n\right).
$$
The inequalities are:


*

*Fatou's Lemma

*A real analysis proposition

*Reverse Fatou's Lemma

