Prove that a smiple graph with at least two vertices would have two or more vertices of the same degree.
Realize that if a simple graph has n vertices then the maximum degree of any vertex is n-1. Thus suppose that we have a simple graph G with a degree sequence of <0,1,2,…,n-1>, that is all the n-vertices are of different degree. Now remove the vertex of degree 0. We now have a new graph G’ with (n-1) vertices and the same number of edges. But it has a vertex of degree (n-1). But that is impossible; so the original graph is impossible. Thus the graph G would have to have at least two vertices of the same degree.