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.