Thermal equilibrium in reversible processes In whatever resources I have consulted to study reversible thermodynamic processes, a common statement seems to be :

The system and surroundings (if the boundary allows) are in mechanical ($P_{sys}=P_{surr})$ and thermal $(T_{sys}=T_{surr})$ equilibrium at all instants.

However, in various answers on the Chemistry stack exchange, and in physical chemistry by peter atkins, it is claimed that the &quot;surrounding&quot; is an infinite reservoir: so the temperature can be assumed to be constant.
1.Is this not contradictory to the &quot;thermal equilibrium&quot; I mentioned in the beginning?
2.Thermal equilibrium of a system with its surroundings means that the entropy change of the surroundings , should be given by $\int\dfrac{-dq_{sys}}{T_{surr}}$. However, everywhere, this equation is simply given as $\dfrac{-Q_{sys}}{T_{surr}}$, which seems to be explained by the claim that the Temperature of the surroundings doesnt change. This equation, according to me should be applicabe only to Irreversible processes happening quickly. (or to isothermal processes).
so what exactly should we consider: $T(sys)=T(surr)$ at each instant, or $T(surr)$=constant?
In a cases where where the temperature is changing during the reversible process, instead of just using a single reservoir at a constant temperature, what you do is use a continuous sequence of reservoirs, all at slightly different constant temperatures (over the range from the beginning to the end of the process), with a tiny (differential) amount of heat exchanged with each.  So, for each reservoir, you have $$dS_{surr}=\frac{dq_{syst}}{T_{surr}}$$ and, if you add all these differential changes in entropy up, you end up with $$\Delta S_{surr}=-\int{\frac{dq_{syst}}{T_{surr}}}$$