Quickly proving the Andr\'asfai-Erd\H{o}s-S\'os-Theorem

Given an integer $r\text{gs}2$, an important theorem first proved by B. Andr'asfai, P. Erd\H{o}s, and V. T. S'os states that any $K_{r+1}$–free graph on $n$ vertices whose minimum degree is greater than $(3r-4)n\mathrm{/}(3r-1)$ is $r$–colourable, and determines the graphs that are extremal in this context. The purpose of this note is to give an alternative proof of this result using a different idea.