Chromatic index. Proof.

Question

Let $\chi(G) $ denote the chromatic number of $G$.

I need to prove that $$\chi(G)\left(\chi(G) -1\right) \le 2|E|. $$

And now I'm asking for help.

Answer 1

Hint: If colouring is minimal there must be at least one edge between every colouring class.