If $\delta >0$ and $N$ is sufficiently large in terms of $\delta$, and $A\subseteq \{1,\dots ,N\}$ is such that ${\sum }_{a\in A}\frac{1}{a}>\delta \log N$ then must there exist $S\subseteq A$ such that ${\sum }_{n\in S}\frac{1}{n}=1$?