Lemma

If $\mathcal{C}$ is the Shannon-Fano Code for the distribution $q(x^{n})$ and $X^{n}$ has pmf $p(x^{n})$ then we have the following bound for the redundancy $$\begin{align*} \\ \frac{1}{n} D(p\|q)\le R(\mathcal{C}_{n},p)< \frac{1}{n}D(p\|q)+ \frac{1}{n} \end{align*}$$ the $\frac{1}{n}$ becomes a $\frac{2}{n}$ if arithmetic coding is used instead.