Let $(a_n)_{n=1}^\infty$ be a sequence such that $0\leq a_n \leq 1$, $\sum_{n=1}^\infty a_n=1$ and let $card \{a_n: n \in \mathbb{N} \}=\infty$. Let's consider the set $$S=\{ \sum_{n\in I} a_n: I \subset \mathbb{N} \}.$$ It may happens that $S=[0,1]$ for example for $(\frac{1}{2},\frac{1}{2^2},\frac{1}{2^3}...)$, or $(\frac{1}{3}, \frac{1}{3}, \frac{1}{3^2}, \frac{1}{3^2},...)$. Under what condition $S$ contain an interval? Is it then $S=[0,1]$?
Does such a subset has a nonempty interior?
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analysis
metric-spaces
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2The conditions. when you get the whole interval $[0,1]$ for a *non-increasing* sequence are known, see [here](http://math.stackexchange.com/a/81552/8297). – 2012-04-21
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0Thanks for links. I don't know if I understood correctly. W.l.o.g. we may assume that $(a_n)$ is decreasing because series is absolutely convergent. Then $S=[0,1]$ iff $a_n \leq \sum_{i=n+1}^\infty a_i$ for all $n \in N$. – 2012-04-22
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0Yes, I agree with your comment. Unless, for some reason, the original ordering of the sequence $(a_n)$ is important for you, you can reorder it to get a non-increasing sequence. – 2012-04-22