An elementary proof of Brémont's theorem on the convergence of equilibrium measures at temperature zero

We give an elementary proof that if $f$ is a locally constant function on a topologically mixing subshift of finite type, then the equilibrium measures for $tf$ converge as $t\to \mathrm{\infty }$, a result due to Brémont. The proof shows that the limit measure assigns an algebraic number to every cylinder.