Assume we have a Graph $G$ and a minimal spanning tree $T$ with weight $w$. Let $Z$ be a cycle, which visits each vertices at least once.
How can i prove that every cycle $Z$ has at least weight $w$.
Intuitively, it is clear that the minimum spanning tree also has minimum weight. But I don't see how this can be proven formally. Can you turn every cycle $Z$ into a minimal spanning tree $T$?
If you delete any edge from $Z$, you get a path $P$ which still visits each vertex exactly one. Since the edges of $P$ are a subset of the edges of $Z$, the weight of $Z$ is at least the weight of $P$.
A path is a tree, so $P$ is a spanning tree. Since $T$ is a minimum spanning tree, the weight of $P$ is at least the weight of $T$.
We have $\text{weight}(Z) \ge \text{weight}(P) \ge \text{weight}(T) = w.$