We have mutually independent random variables $X_n$ with $P(X_n = 2^n) = P(X_n = -2^n) = \frac12$.
Of course their means $\mu_n = 0$ and variances $\sigma_n^2 = 4^k$. Let $S_n = \sum_1^n X_k$. Clearly the mean $m_n=E(S_n) = 0$ and variance $s_n^2 = E(S_n^2)= \sum_1^n \sigma_k^2 = \frac13(4^{n+1}-1)$.
I have shown that in this case the law of large numbers does not apply, and need to show that CLT does/does not apply.
The only method I know to show that the CLT does apply would be Lindeberg's theorem, which I have tried to apply but I can't seem to get anywhere with. Since $\max_k \sigma_k^2/s_n^2$ does not go to zero, I cannot use Lindeberg's theorem to show that the CLT doesn't apply, so I am lost on how to proceed.
How might I proceeed otherwise?