Show that a simple graph G, where the degree of all vertices are at least half of G's order, has a hamiltonian cycle

Question

Consider a graph G on n vertices (n != 2) , where the degree of any given vertex is at least n/2. I am trying to prove it has a hamiltonian cycle. I tried proving that it has a spanning path, but then I noticed: it doesn't follow immediately from that fact that G has a spanning cycle, I also tried maximality arguments, but failed.