Let $G=(V, E)$ be the following graph:

1. $V=\ell^\infty = $ set of bounded real sequences, with the norm $$\|x\|_\infty = \sup_{n\in\mathbb{N}}|x_n|,$$
2. $E = \big\{\{x,y\}: x,y\in \ell^\infty \text{ and }\|x-y\|_\infty = 1\big\}$.

What can be said about the chromatic number $\chi(G)$?