A local limit theorem for unequally distributed random variables

Let $\xi_1,\dots,\xi_n$ be a sequence of independent random variables. Form another sequence
$$\eta_n=\frac{\xi_1+\dots+\xi_n}{B_n}-A_n.\eqno(1)$$
Suppose that for any $n$ $\xi_n$ has one of $\tau$ absolutely continuous distributions
$$F_1(x),F_2(x),\dots,F_\tau(x)$$
The following assertion is proved.
For the sequence of the densities $p_n(x)$ of the sums (1) to converge uniformly to the density of a limit law for some $B_n>0$, $A_n$ it is necessary and sufficient that
1. $\mathbf P\{\eta_n<x\}\to G(x)$ weakly ($G$ is the limit law).
2. There exists such an $N$ that $p_N(x)$ is bounded.