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Sample a set of N numbers without replacement, each element taken from N different weighted sets

There is a mapping from each possible list $P$ to an interval $[\ell,r)$ contained in $[0,1)$, such that the resulting intervals (if you consider all possible lists) form a partition of $[0,1)$. In ...
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Since $X_1,X_2$ are independent, $$H(X_1+X_2) \geq H(X_1+X_2|X_1) = H(X_2),$$ and similarly $H(X_1+X_2) \geq H(X_1)$. Therefore  H(X_2|X_1+X_2) + H(X_1+X_2) = H(X_2,X_1+X_2) = H(X_1,X_2) \leq \\ H(...