Solution
= Solution
Let $P_i$ be the marginal <probability distribution> of $X_i$. <Subadditivity of information entropy> and the <concavity of information entropy> give
$$
H(X_1^n)\leq\sum_{i=1}^nH(P_i)
\leq nH\!\left(\frac1n\sum_{i=1}^nP_i\right).
$$
For each $a\in A$,
$$
\frac1n\sum_{i=1}^nP_i(a)
=\mathbb P(X_J=a)=\overline P(a),
$$
so $H(X_1^n)\leq nH(\overline P)$.