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

Question

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!

score 3 · Answer 1 · answered Apr 11 '16 at 07:19

3

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?

answered Apr 11 '16 at 07:19

hmakholm left over Monica

286,031

Where do you get that each set is finite? – pancini Apr 11 '16 at 16:44
@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. – pancini 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 – pancini Apr 11 '16 at 16:54
@ElliotG: Are you trying in a roundabout way to assert that I should have said "finite or countable" instead of "countable"? I'm confused, because my answer uses neither of the words "finite" or "countable" about $S\cap (1/n,\infty)$ but instead leaves it to the reader to draw conclusions about it, with whatever terminology he prefers. – hmakholm left over Monica Apr 11 '16 at 16:59
Let us continue this discussion in chat. – pancini Apr 11 '16 at 17:00

score 0 · Answer 2 · edited Apr 13 '17 at 12:21

0

This is strongly related to The sum of an uncountable number of positive numbers. Using this you know that all but countably many elements of S are 0. Since 0 appears at most once you are done.

edited Apr 13 '17 at 12:21

Community

1

answered Apr 12 '16 at 06:37

YannickSSE

620

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

2 Answers2