= Tail finite-sums semigroup
{title2=$K=\bigcap_r\overline{\operatorname{FS}(x_r,x_{r+1},\ldots)}$}
For a sequence of positive integers, the nested closures of its tail <finite-sums sets> have nonempty compact intersection in the <Stone-Čech compactification of the natural numbers>. This intersection is a <subsemigroup>. For a sum $s$ from one tail, all sums supported beyond a chosen finite representation of $s$ lie in that tail translated by $-s$. The <ultrafilter> addition formula then establishes closure under addition. The <Ellis–Numakura lemma> supplies an <idempotent ultrafilter> containing every tail finite-sums set and hence any set containing the full <finite-sums set>.
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