As stated in an earlier section, here is the required proof. I’ve often seen this ‘left as an exercise for the reader’.  Such exercises are fine -once you understand the material but not as part of the explanation. One danger is that those who are less fluent with mathematical techniques are left wondering where all this stuff comes from…

One thing to note about the last section is the potential for confusion in connection with the signs of Qs. My recommended approach is described below.

First, some straightforward, textbook stuff which talks about the Carnot Cycle. This is the cycle of processes by which heat can be turned into work with maximal efficiency. Even these most efficient, idealised, frictionless machines, operating on the Carnot cycle, will have their efficiency limited by the second law.