Erdős problem 312

Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with $\sum_{n \in A} 1/n > K$ there exists some $S \subseteq A$ such that $1 - \exp(-(c*K)) < \sum_{n \in S} 1/n \le 1$?