 Idempotent-ultrafilter proof of Hindman's theorem

Choose an idempotent ultrafilter on the natural numbers $\mathcal{U}$ and a color class $A\in \mathcal{U}$. The setalso belongs to $\mathcal{U}$, and $A^{\ast }-x\in \mathcal{U}$ for every $x\in A^{\ast }$. Recursively choosing each new term from the finitely many required translates of $A^{\ast }$ puts every nonempty finite sum in $A$.