Solution
= Solution
Stationarity gives $H(X_n)=H(X_{n+1})=H(X_1)$. The Markov relation $X_1\to X_n\to X_{n+1}$ and the <data processing inequality for mutual information> imply
$$
I(X_1;X_{n+1})\leq I(X_1;X_n).
$$
Using $H(X_n\mid X_1)=H(X_n)-I(X_1;X_n)$ on both sides gives
$$
H(X_{n+1}\mid X_1)\geq H(X_n\mid X_1).
$$