TITLE: A restricted version of Riemann series theorem: rearrangements with alternating signs
QUESTION [5 upvotes]: If $(a_{n})$ is a conditionally convergent series in real field, then for any real number $\alpha$, there exists a rearrangement $(a_{k_{n}})$ of $(a_{n})$ such that for all even $n$, $a_{k_{n}} \geq 0$, for all odd $n$, $a_{k_{n}} \leq 0$, and $\sum a_{k_{n}} = \alpha$.
This problem boils down to the following problem: can every conditional real series be rearranged to an alternative convergent series. If this is solved, then apply the extensions of Riemann's theorem by Sierpiński, the original problem is done.
I appreciate any suggestion about this problem. Thanks in advance.

REPLY [4 votes]: Here you prescribe in addition the sequence of signs of the rearranged series in the Riemann-Dini theorem to be alternating,  but note that any non-stationary binary sequence of signs does as well. More precisely:

Let $(a_k)_{k\in\mathbb N} $ be an infinitesimal sequence of non-zero real
numbers such that $\sum_{k\ge0}a^+_k=\sum_{k\ge0}a^-_k=+\infty$.
Let $\epsilon\in \{-1,1\}^\mathbb{N}$ be a non-eventually constant
sequence.
Let $\alpha\in\mathbb R \cup\{ \pm\infty\}$.
Then there exists a permutation $\sigma$ of $\mathbb N$ such that
$$\sum_{k=0}^\infty a_{\sigma(k)}=\alpha$$
$$\text{sgn}\,a_{\sigma(k)}=\epsilon_k.$$

To this end: extract a subset  $S\subset\mathbb N$  such that $\sum_{k\in S}a_k$ is absolutely summable and $a_k$ are positive resp. negative for infinitely many $k\in S$ (therefore for infinitely many $k\in \mathbb N\setminus S$ as well, because of the assumption $\sum_{k\ge0}a^+_k=\sum_{k\ge0}a^-_k=+\infty$).
Then do the Riemann-Dini bijection $\tau:\mathbb N\to \mathbb N \setminus S$ relatively to the sequence $(a_k)_{k\in \mathbb N\setminus S }$ and the number $\alpha-\sum_{k\in S}a_k$, namely $$\sum_{k=0}^\infty a_{\tau(k)}=\alpha-\sum_{k\in S}a_k.$$
Finally, insert the coefficients $\{a_k\}_{k\in S}$ in some order into the series $\sum_{k=0}^\infty a_{\tau(k)}$, so as to get a rearrangement with the prescribed final sequence of signs $(\dagger)$. Since $\sum_{k\in S}a_k$ is absolutely summable, the order of the insertion does not affect the convergence and the value of the sum, which is $\alpha$ as wanted.
$(\dagger)$ This can easily be done, for the reason that any non eventually constant binary sequence contains any other non eventually constant binary sequence as a subsequence, in such a way that the complement is also non eventually constant.