How to find spanning tree of a graph that minimizes the maximum edge weight?
What follows is taken from Tsuyoshi Ito's comment. 

If you know Kruskal’s algorithm for the minimum spanning tree, it is an easy exercise to show that the output of Kruskal’s algorithm is a minimum bottleneck spanning tree. (I think that it is easier than showing that the output of Kruskal’s algorithm is a minimum spanning tree.)