Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 224 / 1 / a / Solution

Because $X$ and $Y$ are independent random variables, conditioning on $Y$ merely translates $X$. The conditional entropy under a deterministic change of variables and the fact that conditioning reduces entropy therefore giveInterchanging $X$ and $Y$ similarly gives $H(X+Y)\ge H(Y)$, and henceThe proof applies to countable alphabets whenever the displayed information entropies are finite.