= Solution
Suppose an engine were more efficient than a reversible Carnot engine between the same reservoirs. Couple it to a reversed Carnot engine, scaling the cycles to have equal <work>. With no net external <work>, their combined effect would be to transfer <heat> from the colder reservoir to the hotter one, contradicting the <Second law of thermodynamics>. \b[No such engine can exceed $\eta_C=1-T_c/T_h$.] Equivalently, the cyclic <entropy> inequality gives $-Q_h/T_h+Q_c/T_c\ge0$, hence the same bound.
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