Let $G$ be connected weighted graph and $P$ its tree with smallest weight.
Let $e = (u,v)$ be an edge, that isn't in the tree $T$. Then there is exactly one simple $u \to v$ path in $T$, say $P$.
Let $e'$ be a random edge on that path. I need to prove that the edge $e'$ has a lower weight than $e$. How should I go about doing that?
HINT Use the fact that $T$ is the minimum weight tree to show that the entire path $P$ should have weight less than $e$, and since all weights must be positive, each edge on $P$ has smaller weight than the entire path.