I am looking for a way to prove that :
$$ H(A,B) \leq H(A)+H(B) $$
from the definition of the shannon entropy :
$$H(X)=-\sum_l P(X=x_l) log(P(X=x_l))$$
Is there a very simple way to prove it with the definition only ?
I ended up with the fact that the problem is equivalent to prove that :
$$ \sum_b P(b) ln(P(b)) - \sum_a P(a) \sum_b P(b/a) ln(P(b/a)) \leq 0$$ but I am stuck...
I am looking for a way using definition if it exists because I don't really know anything about information theory, I am mainly focusing on thermodynamic (where as you know entropy is a very important notion).
The chain rule for entropy states $H(A,B) = H(A) + H(B \mid A)$, so it suffices to prove $H(B \mid A) \le H(B)$, which I believe is the last inequality that you have written.
The quantity $H(B) - H(B \mid A)$ is known as the mutual information of $A$ and $B$, denoted $I(A ; B)$. One can prove that mutual information (in fact, any KL-divergence) is nonnegative, using Jensen's inequality.