Let $a_1, a_2, a_3,...$ be a sequence of non-negative real numbers, let $S_1, S_2, S_3,...$ be a sequence (finite or infinite) of disjoint nonempty sets of natural numbers whose union is $\{1,2,3,...\}$ and suppose that for each $i$ such that $S_i$ is infinite the series $\sum_{n\in S_i}a_n$ converges and that if the number of sets $S_1, S_2, S_3,...$ is infinite then the series $\sum_{i=1}^\infty(\sum_{n\in S_i} a_n)$ converges.
Prove that the series $\sum^\infty_{n=1} a_n$ converges.