Solution
= Solution
Let $\sigma^2=\operatorname{Var}(2^{X_1})\le\ell2^b$. If $\sigma=0$ the result is immediate. Otherwise the <central limit theorem> applies to $L$, while the displayed threshold is at least $\alpha\sigma/\sqrt n$. Hence
$$
\boxed{\lim_{n\to\infty}\mathbb P\left(\frac{|L-\ell|}{\ell}\ge\alpha\sqrt{\frac{2^b}{n\ell}}\right)
\le2-2\Phi(\alpha).}
$$