The book Commonly Asked Questions in Thermodynamics states:
When we refer to the passage of the system through a sequence of internal equilibrium states without the establishment of equilibrium with the surroundings this is referred to as a reversible change. An example that combines the concept of reversible change and reversible process will now be considered.
For this example, we define a system as a liquid and a vapor of a substance in equilibrium contained within a cylinder that on one circular end has a rigid immovable wall and on the other end has a piston exerting a pressure equal to the vapor pressure of the fluid at the system temperature. Energy in the form of heat is now applied to the outer surface of the metallic cylinder and the heat flows through the cylinder (owing to the relatively high thermal conductivity), increasing the liquid temperature. This results in further evaporation of the liquid and an increase in the vapor pressure. Work must be done on the piston at constant temperature to maintain the pressure. This change in the system is termed a reversible change. It can only be called a reversible process if the temperature of the substance surrounding the cylinder is at the same temperature as that of the liquid and vapor within the cylinder. This requirement arises because if the temperatures were not equal the heat flow through the walls would not be reversible, and thus, the whole process would not be reversible.
But if the system and surroundings are in fact at the same temperature then why would this process occur at all?
My understanding is that in fact that they are infinitesimally different in temperature so I guess my question is why infinitesimality gets these processes "off the hook" for being irreversible. In other words, why do these infinitesimal changes not correspond to an infinitesimal increase in the entropy of the universe, rather than none at all?