The following is an excerpt of the proof that if a connected graph has all vertices of even degree that is has an euler tour.
We must prove that every connected graph with all vertices of even degree is eulerian. We shall prove this by contradiction. Suppose there exists a connected graph with all vertices of even degree that is not eulerian. Among such graphs, let $G$ be one with the smallest number of edges. Let T be a longest trail in $G$. Since every vertex in $G$ has even degree and $T$ cannot be extended, $T$ must be a closed trail.
My question is regarding to this last statement. Why is it true that $T$ must be a closed trail?