Let G = (V, E) be a weighted, directed graph with weight function w : E → ℝ and no negative-weight cycles. Let s ∈ V be the source vertex, and let G be initialized by INITIALIZE-SINGLE-SOURCE (G, s). Prove that for every vertex ν ∈ V_π, there exists a path from s to ν in G_π and that this property is maintained as an invariant over any sequence of relaxations.