TITLE: Rephrasing a Convergence Result to make use of the Borel-Cantelli Lemma
QUESTION [6 upvotes]: Let $X_n$ be a sequence of non-negative iid random variables. 
Is it true that the condition,
$$\limsup_{n\rightarrow\infty} \frac{X_n}{n} = \infty \text{ almost surely}$$
is equivalent to the condition,
$$\mathbb{P} \Bigg(  \limsup_{n\rightarrow\infty} \Big\{ {\frac{X_n}{n} \geq x} \Big\} \Bigg) = 1 \ \text{ for all } \ x > 0$$
I am wondering because I would like to rephrase the initial condition so as to make use of the Borel Cantelli lemma.

REPLY [3 votes]: EDIT (thanks, Learner):
For Borel-Cantelli, note that $\omega\in \{\text{limsup}_n \frac{X_n}{n}=\infty\}$ iff for every $k> 0$ there is $n\geq 1$ such that $\frac{X_n(\omega)}{n}>k$. So, if $A_k:= \{\frac{X_n}{n}>k\}$, then $\omega \in \cap_{n=1}^\infty \cup_{k=n}^\infty A_k = \text{limsup}_n A_n$ . Thus your initial condition is equivalent to $P(\text{limsup}_n A_n)=1$.