I'm stuck on the following exercise:
Let $\sum_{n=0}^{\infty} a_n$ be a series of real numbers which is conditionally convergent, but not absolutely convergent.
Define the sets $$A_+:=\{n\in\mathbb{N}:a_n \geq 0\}$$ and $$A_-:=\{n\in\mathbb{N}:a_n<0\},$$ thus $$A_+ \bigcup A_-=\mathbb{N}\ \text{and}\ A_+ \bigcap A_-=\emptyset.$$ Then both of the series $\sum_{n\in A_+}a_n$ and $\sum_{n\in A_-} a_n$ are not absolutely convergent.
$\sum_{n\in A_+} a_n$ and $\sum_{n\in A_-} a_n$ can't be both absolutely convergent at the same time is straightforward since it follows that $\sum_{n\in A_+\bigcup A_-}a_n =\sum_{n\in\mathbb{N}} a_n$ is absolutely convergent, a contradiction.
What I haven't been able to do to is exclude the possibility that one of the two converges and the other diverges, i.e. that the remaining two cases:
(1) $\sum_{n\in A_+} a_n$ absolutely convergent, $\sum_{n\in A_-} a_n$ not absolutely convergent;
(2) $\sum_{n\in A_+} a_n$ not absolutely convergent, $\sum_{n\in A_-} a_n$ absolutely convergent;
it leads to a contradiction.
So, I would appreciate any hints about how to carry out this part of the proof.
Best regards,
Lorenzo.