  A new short clear proof of the asymptotics for the number $c_n$ of real roots
of the Bernoulli polynomials $B_n(x)$, as well as for the maximal root $y_n$:
$$y_n=\frac{n}{2\pi e}+\frac{\ln(n)}{4\pi e}+O(1)\quad\text{and}\quad
c_n=\frac{2n}{\pi e}+\frac{\ln(n)}{\pi e}+O(1).$$
