removing and adding one edge results in a new spanning tree

Question

Let $G=(V,E)$ be a graph and $T_i=(V,E_i), i=1,2$ be two spanning trees in $G$ with $E_1\cap E_2=\emptyset$. Let $e_1\in E_1$. Show that there exists $e_2\in E_2$ so that $T:=T_1-e_1+e_2$ is a spanning tree.
My idea:
The edge $e_1$ induces a cut $S$ in $T_1$. If I add $e_1$ to $T_2$ I get a cycle in $T_2$. Therefore there is an edge $e_2\in E_2$ of this cycle which is contained in $S$.
Now I thought about removing $e_1$ from $T_1$ and adding $e_2$ to $T_1$. But I am not sure if I get a spanning tree.
Thanks for any help.

Answer 1

Hint:

The problem that I see in your approach is that knowing that there's a cycle in $T_2$ doesn't tell you much about cycles in $T_1$.

I would suggest that you look at the construction in the other order: Let $e_2$ be an edge of $E_2$ that is not in $E_1$. Adding this edge to $T_1$ would result in a cycle, so there is some other edge $e_1$ in $T_1$ which can be removed to eliminate a cycle. Now, try to work with $T_1-e_1+e_2$.