Whilst studying information theory, I found this problem: Given random vectors $U=(U_1,...,U_k)$ (input) and $V=(V_1,..,V_k)$ (output) such that $U_i$ are independent, show $$\frac{1}{\log(M)}\sum_i I(U_i;V_i)\leq I(U;V)$$ where $M$ is the size of the code.
We have $p(u)=\Pi_i p(u_i)$ and $p(u|c)=\Pi_i p(u_i|c_i)$.
I am able to write $$\sum_i I(U_i;V_i)=\sum_i\left(\sum_j p(u_{i_j})\log(u_{i_j})-\sum_j H(U_i|v_j)p(v_{i_j})\right)$$ and $$I(U;V)=\sum_i p(u_i)\log(p(u_i))-\sum_j H(U|v_j)p(v_j).$$
I have tried to further decompse these two expressions but with no success. I also can't find a way to make $\log(M)$ "appear". Maybe from Fano's inequality? I appreciate any help. Thank you in advance.