MST Proof (Kleinburg & Tordos) Consider the Minimum Spanning Tree Problem on an undirected graph
G = (V, E), with a cost ≥ 0 on each edge, where the costs may not all
be different. If the costs are not all distinct, there can in general be
many distinct minimum-cost solutions. Suppose we are given a spanning
tree T ⊆ E with the guarantee that for every e ∈ T, e belongs to some
minimum-cost spanning tree in G. Can we conclude that T itself must
be a minimum-cost spanning tree in G? Give a proof or a counterexample
with explanation.
Consider the following non-minimum spanning tree.

           1
 +-------------------+
 |                   |
 |                   |
 |                   |
2|←                 →|2
 |                   |
 |                   |
 |         ↓         |
 +-------------------+
           1