SCHWEITZER 1956 Problem 7

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that, with some positive number $C$ , $\sum_{k=1}^{n}k\mid a_k \mid<n C$ ( $n=1,2, \dots$ )

Putting $s_n= a_0 +a_1+\dots+a_n$ , suppose that $\lim_{n \to \infty }(\frac{s_{0}+s_{1}+\dots+s_n}{n+1})= s$ exists. Prove that $\lim_{n \to \infty }(\frac{s_{0}^2+s_{1}^2+\dots+s_n^2}{n+1})= s^2$ (S. 7)