On Uniform Approximation of the Binomial Distribution by Infinitely Divisible Laws

Let $F_p^n(x)$ be an $(n,p)$ –- binomial distribution function and be the set of all infinitely divisible laws. We define
$$\rho\bigl(F_p^n,\mathfrak G\bigr)=\inf_{G\in\mathfrak G}\sup_x\left|F_p^n (x)-G(x)\right|.$$

Then,
$$\sup_{0\leq p\leq1}\rho\left(F_p^n,\mathfrak G\right)<\frac{C_0}{\sqrt n},$$
where $C_0$ is an absolute constant.