Let $(a_n)$ be a decreasing sequence . If $(na_n)$ converges to zero, does the series $(a_n)$ converge?

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I know the converse is true. I feel that this statement is false, I'm trying to find some counter example. Series $(a_n)$ converging implies $(a_n)$ converges to $0$ , and I tried sequences like $1/n$ (since I know $n^{-p}$ converges iff $p<1$) but I don't get any example.

Can you give me a specific counter example or give exact proof the statement in question. Thanks!