= Solution
Choose a most probable symbol $x_*$, and let $B=\mathbf1_{\{X=x_*\}}$. Its <binary entropy> is
$$
H(B)=h_2(\Pi)=-\Pi\log_2\Pi-(1-\Pi)\log_2(1-\Pi).
$$
Since $B$ is a function of $X$, part (a)(ii) gives $H(X)\geq h_2(\Pi)$. On $[1/2,1]$, the concave function $h_2$ lies above the chord joining $(1/2,1)$ and $(1,0)$. That chord has height $2(1-\Pi)$. Therefore
$$
\boxed{H(X)\geq h_2(\Pi)\geq2(1-\Pi).}
$$
The endpoint $\Pi=1$ is included by <continuity> and corresponds to a deterministic source.
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