Suppose we have a connected, undirected graph $G=(V,E)$, with a weight function $w\colon E\to R$ and let $T$ be a minimum spanning tree. Now assume that we remove a particular edge $e$. The exercise is to find a minimum spanning tree to the new graph.
I've thought of an algorithm: Look at the cut determined by $T$ and $e$, and then replace $e$ by the minimal weighted edge crossing that cut (if such an edge exists). But I don't know how to prove this works. Help?