For any $n\ge2$ prove that $H(X_1,X_2,...,X_n)\ge\sum\limits_{i=n}^\mathbb{n}\ H(X_i|X_j , j \neq i)$ I am trying to figure this out and I am stuck. Any ideas?
For any $n\ge2$ prove that $H(X_1,X_2,\ldots,X_n)\ge\sum\limits_{i=1}^\mathbb{n}\ H(X_i\mid X_j , \ j \neq i)$

Hint: Use the chain rule
$$ H(X_1,\ldots,X_n) = H(X_1) + H(X_2|X_1) + H(X_3|X_2,X_1) + \cdots + H(X_n|X_{n-1},\ldots,X_1). $$
