We form the Hamiltonian closure of a graph by constructing a sequence of graphs Gi with G0 = G, and Gi formed from Gi-1 by adding an edge between two nonadjacent vertices whose degree sum is at least nv. When we reach a Gi to which we cannot add such an edge, we call it a Hamiltonian Closure of G. Prove that a Hamiltonian Closure of a simple graph G is Hamiltonian if and only if G is.