$P(\limsup_nX_n\geq x)\geq\limsup_nP(X_n>x)?$

Question

Let $\{X_n\}$ be a sequence of random variables. Does it hold that for every $x\in\mathbb R$ $$P(\limsup_nX_n\geq x)\geq\limsup_nP(X_n>x)?$$ If yes, what can be used to prove this?

Answer 1

As @Math1000 pointed out, this comes from the reverse Fatou's lemma.

$$\limsup_nP(X_n\geq x) = \limsup_n\int 1_{(X_n \geq x)}dP \\ \leq \int \limsup_n 1_{(X_n\geq x)}dP = P(\limsup_n {(X_n\geq x)}).$$

To verify the conditions of the theorem are satisfied, one can simply use $g \equiv 1(\omega)$ as the non-negative integrable function.