Exponential concentration of a weighted mean ($\underset{\tau \downarrow 0}{\lim }\frac{{\sum }_j{a}_j{e}^{a_j/\tau }}{{\sum }_j{e}^{a_j/\tau }}=\underset{j}{\max }{a}_j$)

For finitely many real numbers $a_1,\dots ,a_m$, let $M={\max }_j{a}_j$. Divide numerator and denominator of their exponentially weighted mean by $e^{M/\tau }$. Every term with $a_j<M$ vanishes as $\tau \downarrow 0$, while each maximizer contributes $M$ to the numerator and one to the denominator. Their quotient therefore tends to $M$, even when the maximum is negative or attained more than once. The finiteness of the list is used to pass the limit through each sum.