  We give a short proof of a bound on the list chromatic number of graphs $G$
of maximum degree $\Delta$ where each neighbourhood has density at most $d$,
namely $\chi_\ell(G) \le (1+o(1)) \frac{\Delta}{\ln \frac{\Delta}{d+1}}$ as
$\frac{\Delta}{d+1} \to \infty$. This bound is tight up to an asymptotic factor
$2$, which is the best possible barring a breakthrough in Ramsey theory, and
strengthens results due to Vu, and more recently Davies, P., Kang, and Sereni.
Our proof relies on the first moment method, and adapts a clever counting
argument developed by Rosenfeld in the context of non-repetitive colourings. As
a final touch, we show that our method provides an asymptotically tight lower
bound on the number of colourings of locally sparse graphs.
