In my thermodynamics schoolbook, it is asked to apply the Clausius inequality to a Carnot cycle. According to the book the answer is to evaluate $\oint\frac{\delta Q}{T}$ over the cycle which should yield $$ \oint\frac{\delta Q}{T} = \frac{Q_H}{T_H}+\frac{Q_C}{T_C}=0 $$
- $Q_H$ is the heat transferred from the hot reservoir (positive)
- $T_H$ is the temperature of the hot reservoir
- $Q_C$ is the heat transferred to the cold reservoir (negative)
- $T_C$ is the temperature of the cold reservoir
While it is easy enough to show this is true using $\delta Q =TdS$, my schoolbook doesn't introduce entropy until later. Therefore, my question is how to prove this equality without using entropy ?