= 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> is
$$
H(W\mid B)=\alpha H(U)+(1-\alpha)H(V).
$$
Since conditioning cannot increase <information entropy>,
$$
\boxed{H(W)\geq\alpha H(U)+(1-\alpha)H(V).}
$$
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.
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