Graph Theory - Minimum Spanning Tree (MST) Question

You are right, any MST is an economic tree.

You can prove that the maximum cost of an edge in an MST is equal to the minimum cost $c$ such that the graph restricted to edges of weight at most $c$ is connected.

This will imply your proposition.

More details. Let $w : E \to \mathbb{N}$ be the weight function. For $t \in \mathbb{N}$, let $G_t = (V,\{e \in E : w(e) \leq t\}$, and let $t_0$ be minimum such that $G_{t_0}$ is connected. Also let $T \subseteq E$ be a minimum cost spanning tree, and $w_T = \max_{e \in T} w(e)$. We want to prove $w_t = t_0$.

To prove that $w_T \geq t_0$, you just have to prove that $G_{w_T}$ is connected, which should be easy, then it follows by the choice of $t_0$.

To prove that $w_T \leq t_0$, we need this proposition:

Proposition: for any minimum spanning tree $T \subseteq E$ and any $e \in T$, there is a cut $\delta(X)$ in $G$ such that $e \in \delta(X)$ and $w(e) = \min_{e' \in \delta(X)} w(e')$.

You can choose for $X$ one of the two connected component of $T \setminus e$.