# An argument for principle of maximum entropy which I don't understand

While having a look at a book on statistical physics (Statistical Physics and Protein Folding), I came across an argument for the principal of maximum entropy which I don't understand. It goes as follows. Assume there two subsystems $S_1$ and $S_2$ in contact with each other with total energy $E=E_1 + E_2$. Total energy is constant but the subsystems are allowed to transfer energy to each other. Assume that the we have a experimental resolution $\Delta$ for energy levels of the system AND $E_0$ is the minimum energy the subsystems can achieve. Let $\Gamma_1(E_1)$ and $\Gamma_2(E_2)=\Gamma_2(E-E_1)$ be the total number of accessible states in each subsystem with given energy $E_1$. Then the total number of states in the whole system is $\sum_{E_0 < E_1 < E}\Gamma_1(E_1)\Gamma_2(E-E_1)$ and the total entropy is then $S(E)= k_B \ln(\sum_{E_0 < E_1 < E}\Gamma_1(E_1)\Gamma_2(E-E_1))$. Now assume that in the summation the maximum term is achieved at $E_1 = \bar{E}_1$. Then this summation (i.e the total entropy) has the lower bound $$k_B \ln(\Gamma_1(\bar{E}_1)\Gamma_2(E-\bar{E}_1)),$$ and the upper bound $$k_B \ln(\Gamma_1(\bar{E}_1)\Gamma_2(E-\bar{E}_1)) + k_B \ln(\frac{E}{\Delta}).$$ Then the book goes on saying that in a macroscopic system of N particles we expect $S$ and $E$ both to be of order $N$. Therefore we can write $$S(E) = k_b \ln\Gamma_1(\bar{E}_1) + k_b \ln\Gamma_2(E-\bar{E}_1) +O(\ln(\frac{N}{\Delta}))$$ So then if we "neglect" the last term then $$S(E) = S(\bar{E}_1) + S(\bar{E}_2)$$ with $\bar{E}_2 = E -\bar{E}_1$. Now the principle of maximum entropy says entropy of an isolated system never decreases. I don't quite understand how the equation above gives us this since I don't see how the equation above says anything about subsystems $S_1$ or $S_2$. I don't think it implies that the system $S_1$ prefers the energy $\bar{E}_1$ and moreover as our ability to make more precise measurement increases and $\Delta \rightarrow 0$ the error term $\ln$ grows and so it sort of says that the above equality is susceptible to fluctuations. Can somebody clarify these points?