When an irreversible cooling (or heating) becomes reversible

In order to make clear the meaning of `reversible process', two cases of irreversible cooling (heating) are analysed. It is assumed that a body is cooled (heated) from an initial temperature T_i to a final one T_f by being placed in contact with a set of n heat reservoirs with temperatures diminishing (increasing) in a geometric or arithmetic progression. The total entropy change ΔS_total, the body entropy change ΔS_body and the change in the reservoir entropy ΔS_res are evaluated. It is explicitly shown that ΔS_total > 0 for any finite n, but as the number of heat reservoirs goes to infinity, the total entropy change goes to zero, i.e. the process becomes reversible.