 Solution

Introduce a selector $B$ with Bernoulli distribution of parameter $\alpha$, and construct $W$ by taking its conditional law given $B=1$ to be $p_U$, and its conditional law given $B=0$ to be $p_V$. No prescribed joint law of $U,V$ is needed. Summing over the selector gives the desired mixture probability mass function. The conditional entropy isSince conditioning cannot increase information entropy,This proves the concavity of information entropy. The argument also covers infinite information entropies with the convention that a zero mixing weight contributes zero. At $\alpha =0$ or $1$, the mixture is the corresponding original law and equality is immediate.