Minimum size of a sufficient set for an IID source ($|S_n|\ge \frac{1-ϵ}{2}{2}^{n(H-\delta )}$)

For finite-alphabet independent and identically distributed random variables, fixed $0\le ϵ<1$ and $\delta >0$, every epsilon-sufficient set $S_n$ of length-$n$ words satisfies $|S_n|\ge (1-ϵ)2^{n(H-\delta )}/2$ for sufficiently large $n$. Intersect $S_n$ with a typical set whose excluded mass is at most $(1-ϵ)/2$. The intersection retains that much probability and each word has probability at most $2^{-n(H-\delta )}$.