Difference in heat capacity and negligible energy exchange 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?
 A: Let the heat capacity of the bath be $C_0$ and the heat capacity of your test system $C_1$. If we transfer some quantity of heat between the two systems then the changes in temperatures are:
$$\begin{align}
\Delta T_0 &= \frac{Q}{C_0} \\
\Delta T_1 &= -\frac{Q}{C_1}
\end{align}$$
If we take the ratio the heat cancels out and we get:
$$ \Delta T_0 = -\frac{C_1}{C_0} \Delta T_1 $$
If $C_0 \gg C_1$ that is the heat capacity of the bath is much greater than the heat capacity of the test system we find $\Delta T_0 \ll \Delta T_1$ i.e. the change in temperature of the bath is negligible compared to the change in temperature of the test system. This is the justification for ignoring the change in tmeperature of the bath.
A: Heat capacity $C$ is the amount of energy needed to raise the temperature 1 degree (actually 1 Kelvin).


*

*If the object added to the bath has a low $C$, little energy is required from the bath to heat it up to the new temperature.


But "little" is relative.


*

*If the bath has a low $C$ as well, then only a little energy taken away from it will reduce it's temperature significantly.


If both have "a low value" (or "a high value") of $C$, the temperature is not constant. Their temperatures will just reduce and rise, respectively, and meet in the middle.
Low or high doesn't really matter - only the difference between their $C$'s is important, since having an object with low $C$ in a bath of high $C$ (the $C$ being low in comparison to the bath's $C$) will cause a much smaller amount of energy to be absorbed from the bath, than what will be needed to change the bath temperature significantly.
The size (mass) of object and bath is included in the heat capacity, $C=mc$, and is thus taken into account ($c$ being the specific heat capacity or the heat capacity per kg). Naturally all this depends on how big a temperature change that is needed - adding a white-hot iron bar into a water bath might cause temperature fall, while adding a less-hot iron bar into it might not.
