We are asked to show if $X_i$ are independent but not i.d. and where $X_i$ are all bounded with $S_n = \sum_n X_i$ and $s^2_n = Var(S_n)$ with $s^2_n \to \inf$ then $\frac{S_n}{s_n}$ has a central limit thus converging in distribution to $N(0,1)$.
I'm trying this with either Lindeburg or Lyapunov with the following statement:
if $X_i$ are bounded then $|X_i| <= M$ for some M and thus $X^2$ is bounded too by $M^2$. The Lindeburg condition is:
$\lim_{n \to \inf}$ $\frac{1}{s^2_n}\sum_n$ $\int_{|X_i| \gt \epsilon*s_n} X^2_i dP$
Since the $X^2_i$ are bounded then they can be taken out of the integral and thus the sum becomes a sum of constants divided by inf $\to$ 0 and thus the Lindeburg condition holds which means a central limit exists.