The proof has been given here and here. A version of the proof found in Reinhard Diestel's Graph Theory edition 2000 gives the following:
Proof. A graph on $V$ has $\frac{1}{2} \sum_{v\in V} d(v)$ edges, so $\sum d(v)$ is an even number. QED.
Given that I understand how the more elaborated proofs given by others proved in the links above, how does the proof given in the aforementioned graph theory book correlate to the more elaborated proofs given? Is it simply making a lot of assumptions about the reader, which makes it so short?
Edit: Note, to my understanding so far, is that it is saying for a given graph $G = (V, E)$, we know that the sum of degrees $\sum_{v\in V} d(v) = 2\cdot |E| \to |E| = \frac{1}{2}\sum_{v\in V} d(v)$. Times it by $2$ shows that the number of degrees is even. So I guess it is assuming most of the latter of half of the more elaborated explanations given by others?
Any help is great! Thanks!
