= Solution
The <chain rule for conditional entropy>[chain rule for entropy], with $X_r$ placed last, gives
$$
H_n=H(X_{-r})+H(X_r\mid X_{-r}).
$$
Since $X_{-r}$ includes all the preceding coordinates, conditioning monotonicity gives
$$
H(X_r\mid X_{-r})\leq H(X_r\mid X_1,\ldots,X_{r-1}).
$$
This proves the first assertion for every $r$, including $r=1$. Sum over $r$: the conditional terms on the right sum to $H_n$ by the ordinary chain rule. Thus
$$
nH_n\leq\sum_rH(X_{-r})+H_n,
\qquad
\boxed{H_n\leq\frac1{n-1}\sum_rH(X_{-r})}.
$$
This is <Han's entropy inequality>.
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