"Prove that if H and K are abelian subgroups of a group G, then the intersection of H and K is a normal subgroup of <H,K>"

Just to be clear a normal subgroup of G is a subgroup where its elements multiplied by an element of G is an element in the normal subgroup, and <H,K> is the group where its elements are linear combinations of the elements of H and the elements of K, correct? I ask this just to be sure I understand the question, but I still don't know how to do this proof?