I'm looking at these two problems:
Ex 5.5.1 Suppose that $G$ is a connected graph, and that every spanning tree contains edge $e$. Show that $e$ is a bridge.
Ex 5.5.2 Show that every edge in a tree is a bridge.
We call an edge in a graph $G$ a bridge if the removal of the edge would disconnect $G$.
The argument for both problems seems kind of similar. As we talk about acyclic, connected graphs - namely trees - in both questions.
If we define that every two nodes in a tree are joined by a unique path as a lemma, we should be able to use the same proof for both problems:
In a tree $T$, there is a unique path $P$ between any two vertices $u, v$. Since $P$ is unique, removing any edge $e$ from $P$ necessarily disconnects the components containing $u$ and $v$. Hence $e$ must be a bridge, as its removal disconnects $T$.
Does this make sense or did I think too simple?