As printed the double intersection is not a tail event; read as $\liminf A_n$ it is

Let $\{X_n\}$ be a sequence of independent random variables. Define $A_n=[X_n\ge n^{-1}]$. Let $E=\bigcap_{n=1}^\infty\bigcap_{k=n}^\infty A_k$. Then which of the following is always correct?