Let G be a connected graph. Let p1 and p2 be two paths in G.
1) Prove that if p1 and p2 have no vertex in common, then there exists a path p3 with its vertex in p1 and its last vertex in p2 and any remaining vertices that are not in vertices of p1 and p2.
2) Prove that if p1 and p2 are two longest paths of G, then they have a vertex in common. You may assume 1).