I am reading the paper "chain independence and common information" (<http://ttic.uchicago.edu/~yury/papers/independ.pdf>). In this paper, an inequality is used several times (without proof) which looks interesting to me. I would appreciate if anybody can help me prove it.
The inequality is as follows: For random variables $X$, $Y$, $Z$ we have 
$$H(Z)\leq H(Z|X) + H(Z|Y) + I(X;Y)$$ where $H(\cdot)$ is the entropy and $I(\cdot;\cdot)$ is mutual information. 


As far as I see in the paper, there is no restriction on the structure of random variables $X,Y$ and $Z$, so they are arbitrary. (Look at the first page or second page of the paper in the proof of Theorem I)



Inequality in information theory
The answer is as follows. We need to show that 


$$I(Z; X)-I(X; Y)\leq H(Z|Y).$$ The left hand side of this is simplified as $H(X|Y)-H(X|Z)$, so we need to show that $H(X|Y)-H(X|Z)\leq H(Z|Y)$. Since conditioning reduces the entropy we have 
$$H(X|Y)-H(X|Z)\leq H(X|Y)-H(X|Y,Z)=I(X;Z|Y)\leq H(Z|Y).$$


