limsup in probability

Question

Consider a sequence of $n$ i.i.d. random variables $X_n$. We have the inequality: $$P(\liminf{X_n\leq x})\leq \liminf P({X_n\leq x})\leq \limsup P({X_n\leq x})\leq P(\limsup{X_n≤x})$$

Is there any way to prove (or any conditions under which) that:

$$\log P({X_n ≤x}) \leq \limsup \frac 1n P({X_n \leq x})$$

Answer 1

$\log P(X_n \leq x) \leq 0$ and $ \lim \sup \frac 1 n P(X_n \leq x)=0$ so the inequality is always true.