Thermal efficiency: reversible and irreversible machines

[DavideGenoa]

In the proof of Clausius inequality $\oint\frac{\delta Q}{T}<0$ for an irreversible cycle, I always find the fact that the thermal efficiency of an irreversible machine is **stricly less** than the thermal efficiency of a reversible machine, both operating between temperatures $T_H$ and $T_C$.
Nevertheless my book and all the resources that I have found prove by reductio ad absurdum that, between the temperatures $T_H$ and $T_C$, the thermal efficiency of an irreversible machine is (only) **less or equal** to the thermal efficiency of a reversible machine because, if it were strictly greater, the positive work done by an irreversible thermal machinee could be used to activate a reversible machine used as a chiller, and the resulting composed machine would produce a flow of heat from a cold source at the temperature $T_C$ to a hot one at the temperature $T_H$, violating the second principle of thermodynamics.

How can it be proved that the thermal efficiency of an irreversible machine is strictly less than the thermal efficiency of a reversible machine operating between the same two temperatures $T_C$ and $T_F$?
$\infty$ thanks for any answer!

 

[mfb]

If you have machine A exactly at the ideal efficiency, you can use another ideal machine B to revert the system to the previous state. Or just run A backwards. Your irreversible process is now part of a reversible process => contradiction

 

[DavideGenoa]

Brilliant answer! Thank you very much!