central limit theorem for uniformly bounded random variable

Question

Can someone provide some help in this. Thank you. \ Let the $X_{\ell}$ 's be independent uniformly bounded real random variables. Let $\mu_{\ell}=\mathbb{E}\left(X_{\ell}\right)$, and $\sigma_{\ell}^2=\operatorname{Var}\left(X_{\ell}\right)$ satisfy $c_1<\sigma_{\ell}^2$ for some $c_1$ which does not depend on $\ell$. State and prove a central limit theorem for $\sum_{\ell=1}^n X_{\ell}$. Are we supposed to use characteristics function and frame it in the same way as we did for central limit theorem with iid.