Supposes (G, *) is a group, and H is a non-empty finite subset of G that is closed under *. Prove that H is a subgroup of G. Give an example to show that this is not true without the finite hypothesis.

To prove that H is a subgroup I need to show that H has an inverse and an identity. Since * is associative in G, then * is associative in H, true? How can I show H has an inverse and an identity? Also, how can I find an example that makes this not true?