Let $H$ be a graph with each edge having an edge weight such that no 2 edges have the same weight. Consider $T$ as the minimum spanning tree of $H$. For any vertex $v$ in $H$ consider $E_v$ to be set of all edges that are incident with vertex $v$. Prove that for any vertex $v$ in the graph $H$, the edge with lightest weight in $E_v$ is an edge in $T$.
Suppose the edge with lightest weight of $E_v$ is not an edge in $T$.
Now I get a bit stuck because I know that the justification should be that we can form a more minimum tree, but I am not sure how to justify that exactly?
When you add the lowest weight edge from $E_v$ to $T$ you formed a cycle. What happens if you take out the edge of maximum weight in this cycle after adding the edge. Do you get a new tree ? And the weight of this new tree is it smaller then the weight of $T$ ?