Exercise C.4.7

$\star$ Show that for $0 \le k \le n$,

$$ b(k;n,1/2) \le 2^{nH(k/n)-n} $$

where $H(x)$ is the entropy function (C.7)

$$ b(k;n,1/2) = \binom{n}{k}\frac{1}{2^n} = \binom{n}{n\frac{k}{n}}\frac{1}{2^n} \le \frac{2^{nH(k/n)}}{2^n} = 2^{nH(k/n) - n} $$