Prove that if there exists a chain of subgroups such that and each is simple then is simple.
Suppose we have such a chain, and let be a normal subgroup. Note that is a normal subgroup for each . Since each is simple, we have .
Suppose for all . Then . Hence .
Suppose now that for some . Note that if for some , then . Thus if we let be minimal such that , then by induction for all . Then . Thus .
Since every normal subgroup of is trivial, is simple.