Prove that the law of large numbers may be applied to a sequence of independent random variables \(\left\{\xi_{n}\right\}\) such that

\[ \mathbf{P}\left\{\xi_{n}=n^{x}\right\}=\mathbf{P}\left\{\xi_{n}=-n^{x}\right\}=\frac{1}{2} \]

if and only if \(\alpha<\frac{1}{2}\).