Parts a and b will be regarding undirected graphs, and graph with no self loops. Context: Rooted tree X = (V,E,r) is a tree (V,E) with a vertex which is called the root of the tree. A rooted path in a rooted tree T = (V, E, r) is a simple path whose first vertex is the root r. Proof questions: a) Prove that there exists a rooted tree X = (V, E, r), that has arbitrarily long rooted paths, but does not have an infinite rooted path.

b) Let X = (V, E, r) be a rooted tree where every vertex has a finite degree. Prove that if X has arbitrarily long rooted paths, then it has an infinite rooted path. Note: Asked this question before, and got an AI answer. So if you use AI I will know, and you will be reported. If you know the answer please answer in a written response.