Rearranging a series of nonnegative terms

Question

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.

Answer 1

It suffices to show that the sequence $s_m$ of partial sums is bounded ($s_m=\sum_{n=1}^m$). Let $M=\sum_{i=1}^\infty(\sum_{n\in S_i}a_n)$. Now, let us show that $s_m\leq M$ for all $n\in\mathbb N$. For $m\in\mathbb N$, pick $N$ so that $\{a_1,...,a_m\}\subseteq \bigcup_{i=1}^N S_i$. It follows that $$s_m\leq\sum_{i=1}^N(\sum_{n\in S_i}a_N)\leq M$$