TITLE: On the set of the sub-sums of a given series
QUESTION [19 upvotes]: Choose a sequence $(x_n)_{n\in\mathbb N}$  of nonnegative real numbers with finite sum $x=\sum\limits_{n\in\mathbb N}x_n$ and consider the set $X=\{x_I\mid I\subseteq \mathbb N\}$ where, for every $I\subseteq \mathbb N$,  $x_I=\sum\limits_{n\in I}x_n$. Thus, $X\subseteq[0,x]$.

Question: Can the set $X$ fail to be closed?

A motivation from probability theory is explained there. Note that $X=[0,x]$ if $x_n=1/2^n$ and that $X\neq [0,x]$ if $x_n=a^n$ for some $a$ in $(0,1/2)$ (yielding measure zero Cantor sets), but that in all these cases $X$ is closed.

REPLY [16 votes]: The range of an $n$-dimensional vector measure is always closed. See the paper (and references therein)

P. R. Halmos (1948), The range of a vector measure, Bull. Amer. Math. Soc. 54, 416–421. 

So the set will always be closed.