Is the following statement about Euclidean MSTs true, and if so could someone help me with a proof? Between any two nodes, the EMST minimizes the maximum edge cost of any edge required to traverse from one node to the other.
Edited Statement: Show that in a Euclidean MST-- found by Kruskal's algorithm, given any path between two points, there cannot have been any other path between those two points that used an edge of a smaller cost than any edge used in the original path.