Convergence of $\sum_{n=1}^\infty \frac{a_n}{n}$ with $\lim(a_n)=0$. Is it true that if $(a_n)_{n=1}^\infty$ is any sequence of positive real numbers such that $$\lim_{n\to\infty}(a_n)=0$$
then,
$$\sum_{n=1}^\infty \frac{a_n}{n}$$
converges?
If yes, how to prove it?

No: take $a_n:=\frac 1{\log n}$, then 
$$\sum_{j=2}^{N-1}\frac 1{j\log j}\leqslant \int_2^N\frac 1{t\log t}dt=[\log(\log t)]_2^N\geqslant \frac{\log \log N}2$$
