all prefetching scheme $\phi$ for large $F$ for any $\epsilon>0$, we have $$\begin{aligned}
                R^*\geq&\sup_{\epsilon>0} \limsup_{F\rightarrow\infty}\min_{\boldsymbol{\phi}}\nonumber\\&\{ \frac{1}{F}(\sum_{k=1}^{\min\{N,K\}} H^*(W_{k}|Z_{\{1,...,k\}}, W_{\{1,...,{k-1}\}}))\nonumber\\&- \min\{N,K\}(\frac{1}{F}+\epsilon)\}\nonumber\\
                =&\sup_{\epsilon>0} \limsup_{F\rightarrow\infty}\min_{\boldsymbol{\phi}}\nonumber\\&\{ \frac{1}{F}(\sum_{k=1}^{\min\{N,K\}} H^*(W_{k}|Z_{\{1,...,k\}}, W_{\{1,...,{k-1}\}})\}\nonumber\\
                \geq& \inf_{F\in\mathbb{N_+}}\min_{\boldsymbol{\phi}}\nonumber\\&\{ \frac{1}{F}(\sum_{k=1}^{\min\{N,K\}} H^*(W_{k}|Z_{\{1,...,k\}}, W_{\{1,...,{k-1}\}})\}. 
                \end{aligned}$$ Now we have derived a symmetrized version of the converse bound. To simplify the discussion, we define $R_{\textup{A}}(F,\boldsymbol{\phi})=\frac{1}{F}\sum\limits_{k=1}^{\min\{N,K\}} H^*(W_{k}|Z_{\{1,...,k\}}, W_{\{1,...,{k-1}\}})$. Consequently, $$\begin{aligned}
                R^*&\geq \inf_{F\in\mathbb{N_+}}\min_{\boldsymbol{\phi}} R_{\textup{A}}(F,\boldsymbol{\phi}). 
                \end{aligned}$$

For Step $2$, as mentioned previously in this proof, to derive the converse bound presented in Theorem \[th:general\], we aim to minimize the symmetrized converse $R_{\textup{A}}(F,\boldsymbol{\phi})$ over all prefetching scheme $\boldsymbol{\phi}$. Moreover, we need to prove that it is no less than the RHS of (\[eq:th2\]) for any parameters $s$ and $\alpha$. We present the following lemma, which essentially solves this problem.

\[lemma:stg\] For any parameters $s\in\{1,...,\min\{N,K\}\}$, $\alpha\in[0,1]$, and any prefetching scheme $\boldsymbol{\phi}$, we have $$\begin{aligned}
\label{ineq:p2l}
     R_{\textup{A}}(F,\boldsymbol{\phi})\geq s-1+\alpha-\frac{s(s-1)-\ell(\ell-1)+2\alpha s}{2(N-\ell+1)} M,\end{aligned}$$ where $\ell\in\{1,...,s\}$ is the minimum value such that $${\frac{s(s-1)-\ell(\ell-1)}{2}+\alpha s}\leq  (N-\ell+1)\ell.$$

The proof of Lemma \[lemma:stg\] can be found in Appendix \[app:plstg\]. Note that the lower bound in the above lemma is identical to the converse in Theorem \[th:general\]. Assuming its correctness, then given any $s$ and $\alpha$, we can bound $R^*$ as follows: $$\begin{aligned}
                R^*&\geq \inf_{F\in\mathbb{N_+}}\min_{\boldsymbol{\phi}}R^*_{\textup{A}}(F,\phi) \nonumber\\
                &\geq (s-1+\alpha)- \frac{s(s-1)-\ell(\ell-1)+2\alpha s}{2(N-\ell+1)}M.
                \end{aligned}$$ 