Finitary alternating groups are simple

Definition
Let $$S$$ be an infinite set. The fact about::finitary alternating group on $$S$$, i.e., the group of even finitary permutations on $$S$$ under composition, is a simple group.

Facts used

 * 1) uses::Alternating groups are simple: The alternating group on a finite set is simple when the set has at least $$5$$ elements.
 * 2) uses::Simplicity is directed union-closed: A directed union of simple subgroups is simple.

Proof
The proof follows from facts (1) and (2), and the observation that the finitary alternating group on an infinite set is the directed union of alternating groups on all finite subsets of size at least $$5$$.