I use • to denote function composition

Prove: Suppose f:S→S. If f•f is surjective then f is surjective.

Proof:

Note that f•f is a function from S to S. Suppose f•f is surjective. Let s any element in S. Then there exists some t in S s.t. f•f (t) = s. Thus f(f(t)) = s. Therefore f(t) is an element in S which is taken by f onto s. QED