Let G$=($V$,$E$)$ be an undirected connected graph with distinct, positive edge weights. Let T be a spanning tree of G$.$ We define the bottleneck edge of T to be the edge of T with the greatest cost. We call a spanning tree T of G a minimum$-$bottleneck spanning tree if there is no spanning tree T$\text{'}$ of G with a lower cost bottleneck edge. Prove that a minimum spanning tree of G is also a minimum$-$bottleneck spanning tree a of G$.$