I would like to prove
$$H(X,Y) \le H(X) + H(Y),$$
with $H$ being the entropy of the random variables $X, Y$. I know that I have to use the Gibbs' inequality, but I don't see how. Using the definition of the entropy, it is left to show that
$$-\sum_{x \in img(X), y \in img(Y)} p(X = x, Y = y) \log p(X = x, Y = y) \le -\sum_{x \in img(X)} p(X = x) \log p(X = x) - \sum_{y \in img(Y)} p(Y = y) \log p(Y = y).$$
In order to use Gibb's inequality, I would have to find a proper sum that equals $\sum_{x \in img(X), y \in img(Y)} p(X = x, Y = y)$, but I don't have any idea for that.