Let us say that a set $A \subseteq \mathbb N$ sends a series $\sum_{n \in \mathbb N}a_n$ of real numbers *to infinity* if the subseries $\sum_{n \in A}a_n$ sums either to $\infty$ or to $-\infty$.

Given four or more conditionally convergent series, is there a single $A \subseteq \mathbb N$ that sends them all to infinity?

Notice that I do not require that all of the subseries sum to $\infty$, or that all of them sum to $-\infty$. This may well be impossible, even for just two series $\sum_{n \in \mathbb N}a_n$ and $\sum_{n \in \mathbb N}b_n$, since we may have $a_n = -b_n$ for all $n$. However, we do require that each subseries is made to diverge to either $\infty$ or $-\infty$, and not merely to diverge by oscillation. (If divergence by oscillation is allowed, then the answer to my question becomes a relatively easy *yes*.)

For one series this is trivial: let $A$ be the set of indices of the positive terms.

For two series, the question requires thought, but it is doable. [The idea is that two conditionally convergent series naturally partition $\mathbb N$ into four sets, the partition being determined by where each of the series is positive or non-positive. With a little work, you can prove that either one of these four sets works, or else the union of two of them will.]

For three series, I was able to prove that the answer is yes, but my proof is long, complicated, and frankly . . . ugly. I can't help but think that I'm going about it the wrong way, and that there must be a better approach leading to an easier proof.

For four or more series, I'm stumped.