Guys, I'm reading the proof that a graph $G=(V,E)$ without isolated vertices is an Euler graph if and only if $G$ is not empty, connected, and all vertices have even degree.

At some point in the proof they assume $G$ is connected and each degree is even. Let $W=(v_0,v_1,\dots,v_n)$ be the longest possible walk where each edge occurs at most once. Then we have $v_n=v_0$. Otherwise we would have an odd number of edges in our walk. Because the degree of $v_n$ is even, there is a line $e\ni v_n$ that is not in the walk, say $e=\{v_n,u\}$.