Iid random variables with infinite variance are unbounded

Question

While preparing for an exam I've stumbled upon an exercise I have no idea how to approach:

$X_1, \dots, X_n$ are iid random variables with $E(X_1) = 0$ and $V(X_1)=\infty$

Show that $\mathbb{P}(\{ |X_n| > \sqrt{n} \} \text{ happens infinitely often}) = 1$

So it seems like Borel-Cantelli's lemma should be of help, but I don't see any way to give a bound for $\mathbb{P}(\{ |X_n| > \sqrt{n} \})$. Neither Markov's or Chebyshev's inequalities seem to apply here

Answer 1

$E(X^2)=\int_0^\infty P(X^2>y)dy=\int_0^\infty P(|X|>\sqrt{y})dy\le \sum_0^\infty P(|X|>\sqrt{n})$ is the bound you were looking for, you're on the right track with borel-cantelli.