Assume that the cost function $E(\underline{x})$ takes integer values and let $U(\beta)=\langle E(\underline{x})\rangle_\beta$. Due to the form (13.6) the computation of $U(\beta)$ is essentially equivalent to statistical inference. Assume, furthermore that $\Delta_{\text {max }}= \max \left[E(\underline{x})-E_*\right]$ is bounded by a polynomial in $N$. Show that
$$
0 \leq \frac{\partial U}{\partial T} \leq \frac{1}{T^2} \Delta_{\max }^2|\mathcal{X}|^N e^{-1 / T}
$$
where $T=1 / \beta$. Deduce that, by taking $T=\Theta(1 / N)$, one can obtain $\mid U(\beta)- E_* \mid \leq \varepsilon$ for any fixed $\varepsilon>0$.