minimum spanning tree and minimum heavyweight spanning tree a minimum heavyweight spanning tree is a spanning tree in which the heaviest edge is as light as possible.
Formally,
input : given connected undirected weighted graph, $G$.
output : a spanning tree $T$ for $G$ with property that every spanning tree $T'$ for $G$ has some edge $e'$ such that $w(e')\ge w(e)$  for every edge $e$ in $T$.


provide a greedy algorithm to solved the problem.
is every minimum heavyweight spanning tree a minimum spanning tree ?


first thing comes into my mind was using Kruskal's algorithm with $O(|E| \log |V|)$ running time. If I use kruskal's algorithm for 1, the output will guarantee to be a MST, thus 2 will be true? Can someone verify this for me, thanks in advance
Hints: 


Scaling.
Almost every graph would answer this.