Prove that if G is a simple graph with at least two vertices, then G has two or more vertices of the same degree. Pf/ Assume its true for the base case 2 vertices with no edges or 2 vertices with one edge. Then either the degree is of both is 0 or the degree of both is 1. Therefore both vertices have the same degree. Now assume true for K vertices. WTS True for K+1 vertices. Since its true that K vertices have at least one pair of vertices of the same degree it follows immediately that K+1 vertices have at least one pair of vertices of the same degree. I was also thinking about doing this by contradiction and showing that if all the vertices are different degrees then there must be a loop but I was not sure how to say that.