If $|a_{n+1}-a_n| = \mathcal{O}(1/n)$ and $\frac{1}{N} \sum_{n=1}^N a_n \to 0$ then $a_n \to 0$?

Question

I was recently reading a paper, where the following step was made: suppose that $(a_n)_n$ is a sequence such that $$ |a_{n+1}-a_n| = \mathcal{O}(1/n) $$ and $$ \frac{1}{N} \sum_{n=1}^N a_n \to 0, \quad N \to \infty. $$ According to the author it follows that $a_n \to 0$. Is this always true? It seems like some Tauberian theorem to me.