Let $G = (V, E)$ be a graph with $V = \omega$. Show that if for all $n < \omega$, the graph $G_{n} = (n, E \cap [n]^{2})$ is $k$-colorable, then $G$ is $k$-colorable.
I know how to prove this using ultrafilters, but I am supposed to use König's Infinity Lemma (that infinite finitely branching trees have infinite paths). Can anyone help with this?
Consider the tree having $k$-colourings $\chi\colon [n]\to[k]$ of finite subgraphs $G_n$ as nodes and an edge between a colouring $\chi$ of $G_n$ and a colouring $\chi'$ of $G_{n+1}$ if $\chi=\chi'|_{[n]}$.