Does $\sum_{i=1}^\infty a_i/i < \infty$ imply that $a_i$ has Cesaro mean zero? If $(a_i)_{i=1}^\infty$ is a sequence of positive real numbers such that: 
$$ \sum_{i=1}^\infty \frac{a_i}{i} < \infty. $$
Does this mean that the sequence $(a_i)_{i=1}^\infty$ has Cesaro mean zero? As in
$$ \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n a_i = 0.$$

I just discovered that this result has a name. If anyone wanted a reference, it is called Kronecker's lemma.
