On the set of the sub-sums of a given series
Solution 1:
Consider the compact metric space $(A,d)$ where $A$ is the set of binary sequences with metric $$d(a,b) = \sum_{n=1}^{\infty} \frac{ |a_i - b_i|}{2^n}.$$ Define $f:A\to [0,x]$ by $f(a) = \displaystyle \sum_{n=1}^{\infty} a_n x_n.$ Then $f$ is continuous so $f(A)=X$ is compact.
Solution 2:
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.