Yoshioka (Ch. 3.1) uses the following reasoning to explain why the heat bath can be considered to be at a fixed temperature.
Since the heat capacity of the heat bath (reservoir) is much larger than that of the system under consideration (sample), the amount of energy exchanged between the systems is negligibly small compared with the total energy of the heat bath. Therefore, the heat bath can be considered to be at a fixed temperature.
I understand that if we assume infinite heat capacity for the heat bath, the temperature can be considered fixed; because you'd need an infinite amount of energy to raise the temperature.
Why is the difference in heat capacity relevant for the heat bath being at a fixed temperature?
I assume it is because with negligible energy transfer between the systems, we can assume the temperature of the heat bath to be fixed; because we assume that the total system, seen as a microcanonical ensemble, is isolated.
How can one show rigorously that the energy exchange is negligible because of the difference in heat capacity between the two systems?