Solution
= Solution
The <chain rule for information entropy> and <conditioning reduces entropy> give
$$
H(X,Y)=H(X)+H(Y\mid X)\leq H(X)+H(Y).
$$
Thus <subadditivity of information entropy> holds for two finitely valued <random variables>. Applying this inequality to $(X_1,\ldots,X_{n-1})$ and $X_n$ and then inducting gives
$$
\boxed{H(X_1,\ldots,X_n)\leq\sum_{i=1}^nH(X_i)}.
$$