Inspired by [The set of all limits of sub-series of an absolute convergent series](https://math.stackexchange.com/questions/2062357/the-set-of-all-limits-of-sub-series-of-an-absolute-convergent-series) is the following true?:



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> Let $a\_n$ be a strictly decreasing sequence and $\sum\_1^\infty a\_n=\ell<\infty$ is a convergent series. Is it true to say that the set of all possible value of all subseries $\sum a\_{n\_i}$ of $\sum a\_n$ is whole $[0,\ell]$?
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The set of all possible values of subseries of a convergent positive term series
For convenience define $S\_n = \sum\_{j\le n} a\_j$ and $T\_n = \sum\_{j > n} a\_j$.


Suppose there is $n$ such that $a\_n > T\_n$. Then for $S\_{n-1} + T\_n < x < S\_n$,
$x$ is not the sum of a subseries.


On the other hand, if $a\_n \le T\_n$ for all $n$, then every $x \in [0,\ell]$ is the sum of a subseries. This can be obtained "greedily": include $a\_n$ iff the sum of $a\_n$ and already-included terms $\le x$.


