\[th:general\], we first present the following key lemma, which gives a lower bound on any $\epsilon$-achievable rate given any prefetching scheme.

\[lemma:ga\] Consider a coded caching problem with parameters $N$ and $K$. Given a certain prefetching scheme, for any demand $\boldsymbol{d}$, any $\epsilon$-achievable rate $R$ is lower bounded by [^16] $$\begin{aligned}
                R\geq &\frac{1}{F}\left(\sum_{k=1}^{\min\{N,K\}} H(W_{d_k}|Z_{\{1,...,k\}}, W_{\{d_1,...,d_{k-1}\}})\right)\nonumber\\&- \min\{N,K\}(\frac{1}{F}+\epsilon).
                \end{aligned}$$

The above lemma is developed based on the idea of enhancing the cutset bound, which is further explained in the proof of this lemma in Appendix \[app:lemmaga\]. One can show that this approach strictly improves the compound cutset bound, which was used in most of the prior works. We now continue to prove Theorem \[th:general\] assuming the correctness of Lemma \[lemma:ga\].

The rest of the proof consists of two steps. In Step $1$, we exploit the homogeneity of the problem, and derive a symmetrized version of the converse presented in Lemma \[lemma:ga\]. Then in Step $2$, we derive the converse bound in Theorem \[th:general\], which is independent of the prefetching scheme, by essentially minimize the symmetrized converse over all possible designs.

For Step $1$, we observe that the caching problem proposed in this paper assumes that all users has the same cache size, 