Let G be a connected graph with edge weights w. Suppose T is a minimum spanning tree of G. Let X be any nonempty proper subset V(G). Prove that at least one edge of minimum weight in the cut induced by X is in T.
Sooo this makes sense, because you have to reach every vertex, so you're bound to be able to choose the path with the lowest weight at some point, but I don't understand this "cut induced by X" stuff, and how I'm supposed to use it in the proof.
The Prim algorithm (https://en.wikipedia.org/wiki/Prim%27s_algorithm) is precisely based on this property of minimum spanning trees.
The Proof of Correctness in Section 3 explains it quite well.