I am trying, mostly out of curiosity, to exhibit an infinite countable set $X$ of real, null sequences, such that given a sequence of signs $(s\_n)\in\{-1,1\}^{\mathbb{N}},$ at least one of $(x\_n)\in X$ will lead $\sum s\_nx\_n$ to diverge.

Every construction I have tried has failed, yet I am convinced such an $X$ should exist. Would appreciate any clever ideas, or a proof of the contrary.

**Source:** my question is inspired by [this one](https://math.stackexchange.com/questions/738601/forcing-series-convergence), which treats the case $X$ finite: in that particular case there is no such set because it is possible to construct $(s\_n)$ so that every $(x\_n)\in X$ leads to $\sum s\_nx\_n$ convergent. The case where $X$ has cardinality $|\mathbb{R}|$ is almost trivial, so I am wondering what happens when $X$ is merely countable.