Given a simple1 undirected connected graph $G$ which has two kinds of edges, call them red edges and blue edges, I want to show that if it is possible to construct a spanning tree with exactly $\ell$ blue edges, and it is possible to construct a spanning tree with exactly $m$ blue edges, then for any $\ell\le k\le m$, it is possible to construct a spanning tree with exactly $k$ blue edges.
I don't think this is possible to show using a greedy algorithm / induction. As per the advice of a collaborator, I think you might be able to show this using some properties of cycles, but I'm not really sure where to go. That being said, bonus points will be awarded for any answer which can give a greedy algorithm / proof by induction.
1 - A graph with at most one edge between any unique pair of nodes, and no edge from any node to itself.