Suppose you have a constant $d$ and a sequence $(a\_n)\_n$ such that we have $d < \liminf a\_n$. Note that we do not assume that $a\_n$ is even bounded. I want to prove that there exists a constant $C$ such that $\sum\_{k = 1}^{n-1}\frac{a\_k}{k} \geq C + \sum\_{k = 1}^{n-1} \frac{d}{k}$ holds for all $n \in \mathbb{N}$. I think you have to use a crucial property of the limit inferior but I'm unable to figure it out by myself.

I'd appreciate every help.