Assume the series $\sum u_n$ is Cesaro summable and $\lim_{n\to\infty} nu_n\to 0$. We want to see that the series is (Cauchy) convergent.
Attempt: Let $s_n=\sum_{i=1}^n u_n$ denote the $n$-th partial sum. Then $$s_{n+1}-\frac{s_1+s_2+\dots+s_n}{n}=\frac{u_2+2u_3+3u_4+\dots+nu_{n+1}}{n}$$
Since the sequence $\frac{s_1+s_2+\dots+s_n}{n}$ converges by assumption, it suffices to see that $$\lim_{n\to\infty}\frac{u_2+2u_3+3u_4+\dots+nu_{n+1}}{n}=0$$
How can we finish the proof?