Let $\{X_n\}$ be independent r.v. satisfying the Lindeberg conditon, set $s_n^2=\sum_{i=1}^n Var(X_i)$. Define $\{\xi_n\}$ to be independent r.v. and independence with $\{X_n\}$. The distribution of $\{\xi_n\}$ is $$P(\xi_n=0)=1-\frac1{n^2},~P(|\xi_n|>x)=\frac{1}{n^2}x^{-1}.$$
Show that $$\sum_{i=1}^n\frac{X_i+\xi_i}{s_n}\overset{d}\to \mathcal N(0,1).$$
How can we do to prove the conclusion? Do we need the distribution of $\{\xi_n\}$, or it just states that the random variables have no finite first and second moment?