+\sum_{k=1}^{n-1}(B_k-B_{k+1})+B_n\right]C  {\rho_\alpha}B_n^{-1}\max_{0\le k\le N}\varepsilon^k.\end{aligned}$$ Using $(1-B_1) +\sum_{k=1}^{n-1}(B_k-B_{k+1})+B_n=1$ (see [@lx Equation (3.7)]), the claim follows.\
Since $n^{-\alpha}B_{n-1}^{-1}\le 1/(1-\alpha)$ for $n=1,\dots,N$ (see [@lx Equation (3.21)]) and ${\rho_\alpha}=\Gamma(2-\alpha)\Delta t^{\alpha}$, by we have $$\label{eq:conv_es5}
\|(E^{n},[D_hE^{n}])\|_2\le C\max_{0\le k\le N}\varepsilon^k.$$ Recalling that $E^n$ and $\varepsilon^n$ depends on ${\sigma}$ and that, by -, $\lim_{|{\sigma}|\to 0}\max_{0\le k\le N}\varepsilon^k_{\sigma}=0$, we finally get $$\lim_{|{\sigma}|\to 0}\,\max_{0\le n\le N}\{\|E_{\sigma}^{n}\|_2+\rho_\alpha\|[D_hE_{\sigma}^{n}]\|_2\}=0,$$ which gives the convergence of $m_{\sigma}$ to $m$ in $C(0,T;L^2({\mathbb{T}}^2))$. To get the convergence in $L^2(0,T;H^1({\mathbb{T}}^2))$, it is sufficient to observe that, since $m$ is smooth, the consistency error of the approximation of the fractional derivative is of order ${{\Delta t}}^{2-\alpha}$ (see [@lx Equation (3.3)]). Hence, by , it follows $\lim_{|{\sigma}|\to 0} {{\Delta t}}\sum_{n=1}^N\|[D_hE_s^{n}]\|_2=0$ and therefore the convergence of the gradient of the error.\
In the final step of the proof, we prove the convergence of $u_{\sigma}$ to $u$. By the convergence of $m_{\sigma}$ to $m$ and assumption (F4), we have that ${\tilde U}$ is a solution to a discrete Hamilton-Jacobi-Bellman equation with $M$ in place of ${\tilde M}$, i.e. $$\label{eq:conv_HJ_pert}
\left\{
\begin{array}{ll}
\bar D_{\Delta t}^{\alpha} {\tilde U}_{i,j}^{n} - (\Delta_h {\tilde U}^{n})_{i,j} + g(x_{i,j},[D_h{\tilde U}^{n}]_{i,j}) = f_h[M^{n+1}]_{i,j}+a^n_{i,j},\\[6pt] 
{\tilde U}^{N_T}_{i,j} = u_T(x_{i,j}),
\end{array}
\right.$$ where the perturbation $a^n$ still satisfies . Arguing as in 