I was asked this question in a PhD interview. They didn't say whether $a_n$ is non-increasing or strictly decreasing. I tried various examples such as $a_n=\frac 1{n^p}$ where $p \gt 1$, $a_n=\frac 1{n^2 \ln n}$. Then $n a_n \to 0$.
I couldn't answer this question then.
But when I saw Robert Israel's answer to this question, motivated from it, I defined a sequence $a_n$ as below :
$ a_n = \begin{cases} \frac 1n, & \text{if $n$ is a power of 2} \\ 0, & \text{otherwise.} \end{cases}$
Here, $\sum a_n=\frac 12+ \frac 14+ \frac 18+ \cdots =\frac {\frac 12}{1 -\frac 12}=1$ and $\{n a_n\}$ diverges, because there are infinitely many terms of $0$ and $1$ in it. i.e.
$n a_n = \begin{cases} 1, & \text{if $n$ is a power of 2} \\ 0, & \text{otherwise.} \end{cases}$
So $\{na_n\}$ converges in some cases and diverges in some cases. Am I right?
Note : I have gone through these posts, 1,2, 3 too but they have some extra conditions or specific details.
Thanks.