English: Demonstration of axiom of choice. Each set in the family of sets ${\displaystyle (S_{i})=S_{1},S_{2},S_{3},...}$ (indexed over the natural numbers N) is nonempty and contain a (potentially infinite) number of elements. The axiom of choice allows us to arbitrarily select a single element from each one, forming a corresponding family of elements ${\displaystyle (x_{i})=x_{1},x_{2},x_{3},...}$. In general both collections may be indexed over any set, not just N.