Theorem (Converse Part)

For any $n\ge1$ and source channel code $(f_{n},g_{n})$ we have if $$E[d(X^{n},g_{n}(f_{n}(X^{n})))]\le D$$then the rate $R$ of $(f_{n},g_{n})$ satisfies $$R\ge R(D)$$i.e. the Rate Distortion Function is the lower bound for all rates satisfying the distortion constraint.

Theorem (Achievability of the Rate Distortion Function (Forward Part))

For any $D\ge0$ and $\delta>0$, if $n$ large enough, then $\exists(f_{n},g_{n})$ with distortion $$E[d(X^{n},g_{n}(f_{n}(X^{n})))]<D+\delta$$ and rate $R$ such that $$R<R(D)+\delta$$