A short proof of a theorem on degree sets of graphs

Let G be a finite simple graph with n vertices. The degree of a vertex v, which we write as deg v, is the number of vertices of G adjacent to v. Since there can be at most one edge between any pair of vertices in a simple graph, deg v ≤ n− 1 for each vertex v. One of the most basic results in Graph Theory, which is also easy to prove, is that if we sum the degrees of vertices of a finite simple graph, the sum equals twice the number of edges in the graph; see [1], for instance.