The question says, "Let $G = (V, E)$ be a connected graph that is not Eulerian. Prove that it is possible to add a single vertex to $G$, together with some edges from the new vertex to some old vertices of $G$ so that the new graph is Eulerian."

So far what I've got for my answer is the following, and I want to see if I'm heading in the right direction:

"Let $u, v \in V$. $u$ is connected to $v $. This means that there is a $(u, v)$-path in $G$. Suppose $d(u)$ and $d(v)$ are odd. Let us add a vertex to $V$; namely, $w$. Let us also add some edges incident to $w$ and the vertices of odd degree. Now, $d(u)$ and $d(v)$ are even. $d(w) = 2$, so $d (w)$ is even as well. $\forall v \in V, d(v)$ is even. There are no vertices of odd degree, so $G$ is now Eulerian."

Any help would be appreciated!