Test for the convergence of the sequence $S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$
Hint. As an alternative to a Cesaro-like theorem, one may use the fact that $x \mapsto \dfrac1x$ is decreasing over $[1,\infty)$ to get $$ 0<1+\frac12+\frac13+\cdots+\frac1n<1+\int_1^n\frac{dx}x=1+\log n $$ giving $$ 0<\frac1n\left(1+\frac12+\frac13+\cdots+\frac1n\right)<\frac1n+\frac{\log n}n. $$
You can check here how can this be solved:
$$\lim_{n\to\infty}\frac1n=0\implies \lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac1k=0$$