Need step by step proofs

3: Given a connected and weighted graph G on n vertices. Consider the following algorithm for constructing a minimum weight spanning of G Input: A weighted and connected graph G Initialization: Pick a arbitrary vertex vEV(G) and let Ho C G be the graph that contains only the vertex v Iteration: For every i from 1 to -1, pick the edge ethat has minimum weight over all the edges that have one endpoint in V(H, ) and one endpoint in V (GV(H, 1). Add ei and the endpoint of e in V(G)\V(H-1) to Hi-1 to create the graph HSG Output: T- Hn-1 Prove that this algorithm always produces a minimum weight spanning tree. Hint: You can assume problem 2 is true and you should use it to prove that this algorithm, which is called Prim's algorithm, always produces a minimum weight spanning tree