 Solution

For each $r$, conditioning on all the other coordinates gives no greater conditional entropy than conditioning only on the preceding ones. Summing and using the chain rule for entropy gives directlyEquivalently, the left side is $n{H}_n-{\sum }_{r}H(X_{-r})$, so the same conclusion follows from Han's entropy inequality. The direct proof also covers $n=1$, interpreting conditioning on an empty vector as no conditioning.