Solution
= Solution
For finite $m$, independence and conditional <subadditivity of information entropy> give
$$
\begin{aligned}
\sum_{i=1}^mI(X_i;Z)
&=H(X_1^m)-\sum_iH(X_i\mid Z)\\
&\leq H(X_1^m)-H(X_1^m\mid Z)
=I(X_1^m;Z)\leq H(Z).
\end{aligned}
$$
The partial sums increase because <mutual information> is nonnegative. Taking $m\to\infty$ proves $H(Z)\geq\sum_{i\geq1}I(X_i;Z)$.