Past exam of the information theory course of the University of Oxford / 2023 / 2 / c / i / Solution

For every $x^n\in A^{n,\epsilon }$, the defining typical set constraint impliesLet $S_n=A^{n,\epsilon }\cap B^{n,\epsilon }$. Summing the lower probability mass bound over $S_n$ givesThereforeThe sample mean constraint can only shrink the typical set, so this cardinality upper bound holds for every positive integer $n$.