I am having a really hard time coming up with a proof for this problem. For every finite set $F\subseteq \mathbb{R}$, let $\Sigma(F)$ denote the sum of the numbers in $F.(\Sigma(\emptyset)=0)$. Show that if $S\subseteq (0,+\infty)$ and $\Sigma(F)\leq100$ for every finite set $F \subseteq S$, then $S$ is finite or countable. I can't see why $S$ could be countable but not uncountable. Any suggestions are much appreciated.
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1I think this was asked an uncountable number of times on the site, in several and many variables. – Asaf Karagila Aug 16 '14 at 22:13
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I did search through the questions but did not find a similar one. I'll try looking again. – fishy Aug 16 '14 at 22:15
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2Here are some examples of similar. As for the possibility that $S$ is countable, pick any convergent series whose limit is $\leq 100$. – Asaf Karagila Aug 16 '14 at 22:19
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Thanks for the links. I think I have a better idea on how to approach the problem now. – fishy Aug 16 '14 at 22:28
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Hint: Let $S_n=\{s\in S\mid s>\frac{1}{n}\}$. How many elements can be in $S_n$ if $\sum(F)<100$ for all finite $F\subseteq S$?
Thomas Andrews
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$S=\{1,1/2,1/4,1/8,1/16,\ldots\}$ is an example of a countably infinite set that meets the requirements.
Chris Culter
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@egreg OP said "I can't see why $S$ could be countable but not uncountable", so I gave an example. – Chris Culter Aug 16 '14 at 23:10
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You are misreading that part of the question, @ChrisCulter. The question isn't "are there examples of countably infinite $S$," but why can there exist such infinite sets but not uncountably infinite sets. – Thomas Andrews Aug 16 '14 at 23:45
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@ThomasAndrews Sure, that's the original prompt given to fishy. I guess I tried too hard to interpret fishy's question in a way that isn't a restatement of the original prompt. – Chris Culter Aug 16 '14 at 23:57