CLT for non identically distributed bounded random variables

Question

Let X1, X2, ... be independent, zero-mean, bounded random variables. Let $S_n$ = $X_1 +... + X_n$ with variance $s^2_n$ = Var($S_n$), $s^2_n \rightarrow \infty$. Show that $S_n/s_n$ has a central limit.

I believe that I am supposed to apply either Lindeberg or Lyapunov here but I am struggling on how to go about that

Answer 1

In order to check Lindeberg's condition, note that the term $$ \mathbb E\left[X_j^2\mathbf{1}_{\{\lvert X_j\rvert/s_n>\varepsilon\}}\right] $$ equals to zero if $\lVert X_j\rVert_\infty/s_n\leqslant \varepsilon$.