The random variable series $\{X_k\}$ is as follows: $P(X_k=\pm k)=\frac{1}{2}k^{-\frac{1}{2}}, P(X_k=0)=1-k^{-\frac{1}{2}}$. The variables are independent. I need to prove that the Strong Law of Large Numbers doesn't hold.
It suffices to prove $P\left(\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n X_i=0\right)\neq 1$ according to the SLLN statement, since $E[X_k]=0$. But I have no idea how to continue the proof. Can the limit be directly calculated? Or there are some other ways towards this problem? Thanks for any help.
Edit: based on michh's hint, I drafted a proof. Define $A_n$ as $|X_n/n|>1/2$, we know that $P(A_n)=n^{-1/2}$. The sum of this probability goes to infinity, so according to Borel-Contelli, $P(A_n\text{ i.o })=1$. So we have
$$P\left(\lim_{n \to \infty}\left|\frac{X_1+\cdots+X_n}{n}\right|>0\right)\geq P(A_n \text{ i.o })=1\,.$$
Thus the SLLN doesn't hold. Is this correct?