Prove that for graphs $H$ and $G$, $\chi(G)$ + $\chi(H)$ = $\chi(G+H)$, where $G+H$ is the graph join

Question

I have an idea of how to prove this but I am stuck on how to show part of my argument. I know how to show that $G+H$ is ($m+n$)-colorable, where $\chi(G)=m$ and $\chi(H)=n$, but I am unsure how to show that $G+H$ is not $k$-colorable, for $k<m+n$.

Answer 1

No color in a coloring of $G+H$ can use the same color for a vertex of $G$ and a vertex of $H$, because every vertex of $H$ is connected to every vertex of $G$ in $G+H$. Thus we have to use $\chi(G)+\chi(H)$ colors,$\chi(G)$ to color $G$, and $\chi(H)$ to color H.