# How to show this subset of $\mathbb{R}$ is countable

For a finite set $X$, we write $\sum X$ to be the sum of all the numbers in $X$. Suppose we have a set $S \subseteq \mathbb{R}$ such that $-100\le \sum X \le 100$ for all finite subsets $X \subseteq S$. How to show that $S$ is countable?

I do not really know where to begin. Hints would be nice. Thank you!

For each $n\in\mathbb N$ consider the sets $$S\cap(-\infty,-1/n) \quad\text{and}\quad S\cap(1/n,\infty)$$ There are countably many of these sets, and their union is $S\setminus\{0\}$. How large can each of them be?
• @ElliotG: Imagine what happens $S\cap (1/n,\infty)$ has at least $100n$ elements ... – hmakholm left over Monica Apr 11 '16 at 16:47
• I'm not seeing how we can say anything about $S\cap (1/n,\infty)$. How do we know it isn't uncountable. Even if it is finite, the sum could still be zero. – Elliot G Apr 11 '16 at 16:50
• @ElliotG: How can a sum of $100n$ elements of $(1/n,\infty)$ be zero? – hmakholm left over Monica Apr 11 '16 at 16:51
• Ok I see your point there. Still not seeing how this proves $S\cap (1/n,\infty)$ is countable thought – Elliot G Apr 11 '16 at 16:54