proof verification - If the series $sum_1^infty a_n$ converges, then so does $sum_1^infty frac{{a_n}}{n} $

The problem:

Show that if $\sum_{1}^\infty a_n$ converges and $a_n ≥ 0$ for all $n ≥ 1$, then $\sum_1^\infty \frac{a_n}{n}$
also converges. Is the statement
true without the hypothesis $a_n ≥ 0$ ?

My attempt:

1) $1\geq\frac{1}{n}$, because $a_n\geq0$ we have $a_n\geq\frac{a_n}{n}$
=>if $\sum_{1}^\infty a_n$ converge then $\sum_1^\infty \frac{a_n}{n}$
also converges.

2) if $\sum_{1}^\infty a_n$ converge then partial sums $s_n$ is a bounded sequence, and $\frac{1}{n}$ is decreasing and $\frac{1}{n}\rightarrow0$ from Dirichlet test we have $\sum_1^\infty \frac{a_n}{n}$ converges.

But I'm not sure if this is correct I feel something is wrong.

Answer

It is correct. I would have used Abel's test to justify the convergence of $\sum_{n=1}^\infty\frac{a_n}n$ without the assumption that $(\forall n\in\mathbb N):a_n\geqslant0$. It allows you to deduce that, say, $\sum_{n=1}^\infty\frac{na_n}{n+1}$ also converges.