Let $X$ and $Y$ be discrete random variables taking finitely many values. Define the conditional entropy $H(X \mid Y)$. Suppose $Z$ is another discrete random variable taking values in a finite alphabet, and prove that

$$H(X \mid Y) \leqslant H(X \mid Y, Z)+H(Z)$$

[You may use the equality $H(X, Y)=H(X \mid Y)+H(Y)$ and the inequality $H(X \mid Y) \leqslant$ $H(X) .]$

State and prove Fano's inequality.