Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables with the following distribution:
$$\mathbb{P}(X_n=-n)=\mathbb{P}(X_n=n)=\frac{1}{2\sqrt{n}},\quad \mathbb{P}(X_n=0)=1-\frac{1}{\sqrt{n}}\quad\text{for any}\quad n\in\mathbb{N}.$$
My question is how to show that the weak law of large numbers does not hold for such a sequence? I am only able to prove that the strong law of large numbers fails to hold (by using Borel-Cantelli Lemma). But I have no idea how to proceed with the WLLN, even though I have tried many times. I would be grateful for any hints.