In Shannon's 1948 paper titled "A Mathematical Theory of Communication", in the discussion of the entropy of the joint event, there is no proof for this inequality (or subadditivity of entropy) $$H\left(x,y\right) \le H\left(x\right)+H\left(y\right)$$ in the paper,
however, I found one proof based on Jensen's inequality from this link https://www.cs.princeton.edu/courses/archive/fall11/cos597D/L01.pdf, but I am confused about which convex function is used in this proof, it is said that the convex function is $$\log{\frac{1}{x}}$$ the below is the proof:
$$H\left(x,y\right)-H\left(x\right)-H\left(y\right)$$ $$=\left(-\sum_{i,j}{p\left(i,j\right)\log{p\left(i,j\right)}}\right)-\left(-\sum_{i,j}{p\left(i,j\right)\log{\sum_{j}\ p\left(i,j\right)}}\right)-\left(-\sum_{i,j}{p\left(i,j\right)\log{\sum_{i}\ p\left(i,j\right)}}\right)$$ $$=\left(-\sum_{i,j}{p\left(i,j\right)\log{p\left(i,j\right)}}\right)-\left(-\sum_{i,j}{p\left(i,j\right)\log{p\left(i\right)}}\right)-\left(-\sum_{i,j}{p\left(i,j\right)\log{p\left(j\right)}}\right)$$ $$=\sum_{i,j}\left(p\left(i,j\right)\left(-\log{p\left(i,j\right)+\log{p\left(i\right)+\log{p\left(j\right)}}}\right)\right)$$ $$=\sum_{i,j}\left(p\left(i,j\right)\left(\log{\frac{p\left(i\right)p\left(j\right)}{p\left(i,j\right)}}\right)\right)$$
Here comes the question: $\log{\frac{1}{x}}$ is a convex function, Jensen's inequality is applied, $\log{x}$ is a concave function, the reverse of Jensen's inequality is applied. How to figure out that the function of $$\log{\frac{p\left(i\right)p\left(j\right)}{p\left(i,j\right)}}$$ is the function of $$\log{\frac{1}{x}}$$ or $$\log{x}$$?
If let $x=\frac{p(i)p(j)}{p(i,j)}$, then the function of $\log{\frac{p(i)p(j)}{p(i,j)}}$ would be function $\log{x}$
$$\sum_{i,j}\left(p\left(i,j\right)\left(\log{\frac{p\left(i\right)p\left(j\right)}{p\left(i,j\right)}}\right)\right)\le\log{\left(\sum_{i,j}\frac{p\left(i,j\right)p\left(i\right)p\left(j\right)}{p\left(i,j\right)}\right)\ }$$ $$=\log{\left(\sum_{i,j}\ p\left(i\right)p\left(j\right)\right)}$$ $$=\log{1}=0$$
Then subadditivity of entropy $$H(X,Y)\le H(X)+H(Y)$$ is proven.
If let $x=\frac{p(i,j)}{p(i)p(j)}$ (rather than $x=\frac{p(i)p(j)}{p(i,j)}$), the function of $\log{\frac{p(i)p(j)}{p(i,j)}}$ would be function $\log{\frac{1}{x}}$ then the proof will become
$$\sum_{i,j}\left(p\left(i,j\right)\left(\log{\frac{p\left(i\right)p\left(j\right)}{p\left(i,j\right)}}\right)\right)$$ $$=\sum_{i,j}\left(p\left(i,j\right)\left(\log{\frac{1}{\left(\frac{p\left(i,j\right)}{p\left(i\right)p\left(j\right)}\right)}}\right)\right)\geq\log{\left(\frac{1}{\sum_{i,j}\left(p\left(i,j\right)\cdot\frac{p\left(i,j\right)}{p\left(i\right)p\left(j\right)}\right)}\right)}$$
then we just get $$H(X,Y)-H(X)-H(Y)\ge\log{\left(\frac{1}{\sum_{i,j}\left(p\left(i,j\right)\cdot\frac{p\left(i,j\right)}{p\left(i\right)p\left(j\right)}\right)}\right)}$$
Then subadditivity of entropy $$H(X,Y)\le H(X)+H(Y)$$ cannot be proven yet.