Lower bound for joint entropy

Let
$$H(X_{1})\leqslant H(X_{2})\leqslant,...,\leqslant H(X_{n})$$

I seem to have a problem of finding a lower bond of joint entropy. I proved that upper bound using the chain rule and the fact that information of discrete variables is positive.And I obtained result that

$$H(X_{1}, X_{2},..., X_{n}) \leqslant \sum_{k=1}^{n} H(X_{k})$$

Where should I start? I found result that lower bond is $$max (H(X_{k}))$$?

• Hint: $H(Y|X) = H(X,Y)-H(X) \ge 0$. – Math Lover Apr 12 at 18:15

THe property $$H(X_1,X_2) \ge H(X_1)$$ should be obvious from the interpretation of entropy, as information content (the information provided by $$X_1$$ and $$X_2$$ together cannot be less than that provided by $$X_1$$ alone). And it can also be proved by the chain rule:
$$H(X_1,X_2) = H(X_1)+H(X_2 | X_1) \ge H(X_1)$$ because $$H(X_2 | X_1) \ge 0$$ (as any entropy).
We also can write $$H(X_1,X_2) \ge H(X_2)$$. Hence $$H(X_1,X_2) \ge \max(H(X_1),H(X_2))$$
You only need to generalize this to $$n$$ variables.