Trying to solve a question in my homework and can't tell if my solution is valid or not, it seems too simple, I would like to hear your opinion let $w(e)$ and $w'(e)$ be 2 weight function that is $w,w':E\rightarrow\mathbb{R}$ that maintain $w(e) \leq w'(e) \leq 2w'(e), \forall e \in E$, let $T' = MST(V,E,w')$ and $T = MST(V,E,w)$ prove that $w(T') \leq 2w(T)$ so I thought using induction would be great, and very simple as well, but it seems invalid, any other suggestions? it seems obivious and I cant tell if Im missing something
Thanks!
Your assumption should be $w(e) \leq w'(e) \leq 2w(e)$, for all edges $e \in E(G)$. You currently have $w'(e) \leq 2w'(e)$.
Using that corrected assumption, start with an MST $T$ for $G(V, E, w)$. Now: $$ w(T) = \sum_{e \in E(T)} w(e) \leq \sum_{e \in E(T)} w'(e) \leq \text{TODO} $$
Now use the assumption that $w'(e) \leq 2w(e)$ for all $e \in E(G)$ to complete the part labeled TODO.