We only prove (a) since the proof for (b) is identical. Fix $\delta>0$, then there exists a $T>0$ such that \begin{equation*}
    E(\rho_T) \le E(\pi)+\delta.
\end{equation*} By assumption (iii), there exists some $\varepsilon_0 \geq \varepsilon_1>0$ such that for all $\varepsilon<\varepsilon_1$, we have $T_\varepsilon^* >T_\varepsilon>T$, and 
