Prove $\limsup X_n = 1$ has probability 0

Question

If $X_n$ are i.i.d. random variables U[0,1], is it true that $$ \{\omega : \limsup X_n(\omega) =1\} $$ has probability 0? How would you prove that?

Answer 1

$\forall \epsilon >0, \sum_{n=1}^{\infty}P(X_n >1-\epsilon) = \sum_{n=1}^{\infty}\epsilon = \infty$, thus by Borel-Cantelli lemma, almost surely there are infinitely many $m$ such that $X_m > 1 -\epsilon$, thus $P(\limsup_nX_n(\omega) > 1 -\epsilon) = 1$, let $\epsilon \rightarrow 0$ to get $P(\limsup_nX_n(\omega) = 1)=1$