Is it true that if $(n\Delta s_n)$ is bounded then $(s_n)$ is bounded? Let $(s_n)$ be a sequence of real numbers and $\Delta s_n=s_n-s_{n-1}.$  Is the following always true?

If $(n\Delta s_n)$ is bounded then $(s_n)$ is bounded.


Let $s_n=\sum_{k=1}^n\frac1k$. Then $$Δs_n=\frac1n$$ hence $nΔs_n=1$ for all $n\in \Bbb N$ but $s_n\to +\infty$.
