Could you help me prove that the following sequence of independent random variables satisfy Weak Law of Large Numbers?
\begin{equation} P(x_k=\pm 2^k)=\frac{1}{2^{2k+1}} \end{equation}
\begin{equation} P(x_k=0)=1-\frac{1}{2^{2k}} \end{equation}
I made this,
\begin{equation} E(x_k)=(2^k)(\frac{1}{2^{2k+1}})+(-2^k)(\frac{1}{2^{2k+1}})+(0)\left( 1-\frac{1}{2^{2k}}\right)=0< \infty \end{equation} also,
\begin{equation} P(|\frac{S_n}{n}-0|> \epsilon)=P(|\frac{S_n}{n}|> \epsilon)\leq \frac{V(S_n/n)}{\epsilon^2} \end{equation} also,
\begin{equation} V(\frac{S_n}{n})=\frac{E(x_k^2)}{n} \end{equation}
and then I don't know what to do.
