Suppose G is an undirected weighted graph such that G is not the complete graph but every edge in G has positive weight. Create a complete graph, H, having the same vertex set as G, such that if (v, u) is an edge in G, then (v, u) has the same weight in H as in G, and if (v, u) is not an edge in G, then (v, u) has weight in H equal to the length of a shortest path from v to u in G. Show that the edge weights in H satisfy the triangle inequality.