Tail finite-sums semigroup ($K={\bigcap }_r\overline{\mathrm{FS}(x_r,x_{r+1},\dots )}$)

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.