I would really appreciate it if anyone would validate if my argument ( proof ? ) for the above statement is valid. I am aware of other proofs but the current argument is more of a task in rational/logical thinking and mathematical writing style.
I am aware this is a open ended question but I would also like it if people would comment on my style so that I can improve on it.
Let $G$ be a disconnected finite graph with exactly two vertices, $u$ and $v$, with odd degrees and two components. Suppose that $u$ and $v$ belong to different components of $G$.
Then either of the two possibilities given below must hold such that number of edges in $G$ is a positive integer.
1) There exists at least one other vertex with odd degree in each component of $G$ such that the number of edges in each component is a positive integer. A contradiction with our assumption that $G$ has exactly two vertices with odd degree.
2) There exists an edge joining the two components. Another contradiction, as we have said previously that $G$ is disconnected.