Erdős Problem #316 — Partitioning Reciprocal Sums Below Two

If $A\subseteq\mathbb{N}$ is finite, $0\notin A$, $1\notin A$, and

$$\sum_{n\in A}\frac1n<2,$$

is there always a partition $A=A_1\sqcup A_2$ into disjoint finite subsets such that

$$\sum_{n\in A_i}\frac1n<1\qquad(i=1,2)?$$