I'm trying to solve an exercise from Resnick's book and I'm convinced there is an assumption missing. The following is the exercise's content:
My attempt was to use the definition directly
$$ \mathbb{E} \left[\left( \frac{\sum_{i=1}^n X_i}{n^\alpha}\right)^2\right] = \frac{\mathbb{E}^2\left( \sum_i^nX_i \right) + \text{Var} \left( \sum_i^nX_i \right)}{n^{2\alpha}},$$
the second term is $\leq nc$ so it's limit is 0 when divided by $n^{2\alpha}$ as $n \rightarrow \infty$, but I can't figure out why the first terms also needs to go to 0.
As a counter example: If $\mathbb{P}(X_n = e^n) = 1$, then all the assumptions made are valid, but the result is not true. I think even limiting $\mathbb{E}(X_i)$ would not be enough.
Is there an error on the exercise (assumption like $\mathbb{E}(X_i) = 0$ missing) or there is something wrong with my solution attempt?