Let $\sum_{n=1}^\infty a_n$ a convergent but not absolutely convergent series.
1.1: For $n\in \mathbb{N}$ let $b_n:=max(a_n,0)$, $c_n=-min(a_n,0)$
Show that: $\sum_{n=1}^\infty b_n=\infty$ and $\sum_{n=1}^\infty c_n=\infty$.
1.2: Conclude, that for every $S\in \mathbb{R}$ there is a rearrangement $\sigma:\mathbb{N}\to \mathbb{N}$, so that the series which was rearranged by $\sigma$ converges to $S$, meaning
$\sum_{n=1}^\infty a_{\sigma (n)}=S$.
I'm sorry to repost this, but the original seemed to go nowhere and I have too low of a reputation to comment and I didn't want to use the answer option since I didn't have an answer. Anyway, here is the original one: https://math.stackexchange.com/questions/1333523/rearrangement-of-series-not-absolutely-convergent .
1.1:) Since the series is not absolutely convergent I can see how limit of the sequences with $b_n$ and $c_n$ are infinite since $b_n$ and $c_n$ can't be $0$ because that would consequently mean that it's absolutely convergent. But I don't know how to write and prove this idea mathematically.
1.2:) Like the original author I'm lost here, because I never dealt with rearranging series'. Is there a certain approach to those kind of things?