Contradiction in definition of entropy? I'm studying for my thermodynamics exam and I came across something which really confuses me. 
An infinitesimal change in entropy $ dS_{sys}$  of a system at temperature $T_{sys}$ during a reversible transformation, where $\delta Q_{rev}$ is defined as the heat going in/out the system is given by: $$ dS_{sys} = \frac{\delta Q_{rev}}{T_{sys}} $$
However, there is a statement in my book claiming that: $ dS_{sys} > \frac{\delta Q_{rev}}{T_{surr}} $
My confusion is the following: If an infinitesimal change in entropy of a system at temperature $T_{sys}$ is defined as above, how can the statement $ dS_{sys} > \frac{\delta Q_{rev}}{T_{surr}} $ be true? In order to calculate the change in entropy the path must be reversible, meaning that the temperature of the system is equal to the temperature of the surroundings, i.e. $ T_{sys} = T_{surr} $ otherwise the path isn't reversible. The statement clearly doesn't hold if my reasoning is correct. 
Can someone clarify this to me because I'm really struggling with this. 
 A: For a general transformation between $A$ and $B$, the entropy change can be written:
$$dS_{A \to B} = \frac{dQ_{A \to B}}{T_{\mathrm{surr}}} + dS_{\mathrm{created}},$$
where $dS_{\mathrm{created}} \geq 0$ is the entropy created by irreversible processes.
For a transformation to be reversible, you need $dS_{\mathrm{created}} = 0$ and also $T_{\mathrm{surr}} = T_{\mathrm{sys}}$. In that case, $dS = \frac{dQ}{T_{\mathrm{surr}}} = \frac{dQ}{T_{\mathrm{sys}}}$ so the first inequality does not hold strictly.
However, for real macroscopic processes, $dS_{\mathrm{created}}$ is always $>0$, even by an infinitesimal amount (no transformation is fully reversible). In that case, the inequality becomes strict and $dS > \frac{dQ}{T_{\mathrm{surr}}}$. Of course it is still useful to consider adiabatic processes as they are sometimes a really good approximation to some almost reversible real transformations, and they can also be used for non-reversible process to calculate the change of entropy between two states $A$ and $B$ by considering the corresponding adiabatic path between the initial and the final states (as $dS_{\mathrm{A\to B}}$ does not dépend on the path followed).
