easy to obtain $$\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\int_{0}^{t} S_{t-s} g(u^{N,n}_{s}) e_i \mathrm{d} \beta_{i}^{H,N,n}(s) \rightarrow \sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\int_{0}^{t} S_{t-s}  g(u^{N}_{s}) e_i \mathrm{d} \beta_{i}^{H,N,n}(s)\qquad ~{\rm as}~n \rightarrow \infty,$$ almost surely. Finally, we will obtain $$\begin{aligned}
&& \Big|\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\int_{0}^{t} S_{t-s}  g(u^{N}_{s})e_i d (\beta_{i}^{H,N,n}(s)-\beta_{i}^{H,N}(s))\mathbf{1}_{t}\Big|\cr
&\leq& C \Big(\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}-\beta_{i}^{H,N}\|_{\alpha,0,T}\int_{0}^{t}(r^{-\alpha}+(t-r)^{-\alpha})(1+\|u^{N}\|_{\alpha,r})\mathrm{d}r\Big)\mathbf{1}_{t}\cr
&&+C \Big(\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}-\beta_{i}^{H,N}\|_{\alpha,0,T}\int_{0}^t\Big( \int_{0}^{r}\frac{|u^{N}_r-u^{N}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r\Big)\mathbf{1}_{t}\cr
&\leq& C_{R} \sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}-\beta_{i}^{H,N}\|_{\alpha,0,T}.\end{aligned}$$

Since $\|\beta_{i}^{H,N,n}-\beta_{i}^{H,N}\|_{\alpha,0,T} \rightarrow 0, n \rightarrow \infty$, a.s., then, we have $$\Big|\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\int_{0}^{t} S_{t-s}  g(u^{N}_{s})e_i \mathrm{d} (\beta_{i}^{H,N,n}(s)-\beta_{i}^{H,N}(s))\Big|\rightarrow 0\qquad ~{\rm as}~n \rightarrow \infty,$$ almost surely.

Next, we have $$\begin{aligned}
\mathbb{E} \Big[ \Big|\int_{0}^{t} S_{t-s} (\sigma(u^{N,n}_{s})-\sigma(u^{N}_{s})) \mathrm{d}W_s\Big|^2\mathbf{1}_{t}\Big]
&\leq&  \int_{0}^{t} \mathbb{E} \big[|\sigma(u^{N,n}_{s})-\sigma(u^{N}_{s}))|_{L_2(V)}^2\mathbf{1}_{s}\big] \mathrm{d}s  \cr
        &\leq&  \int_{0}^{t} \mathbb{E} \big[\|u^{N,n}-u^{N}\|_{\alpha,s}^2\mathbf{1}_{s}\big] \mathrm{d}s.\end{aligned}$$

Thus, by Lemma \[lemboundNm\], it is easy to obtain $$\mathbb{E} \Big[ \Big|\int_{0}^{t} S_{t-s} (\sigma(u^{N,n}_{s})-\sigma(u^{N}_{s})) \mathrm{d}W_s\Big|^2\mathbf{1}_{t}\Big] \rightarrow 0\qquad ~{\rm as}~ n\rightarrow \infty,$$ and, consequently $$\int_{0}^{t} S_{t-s} \sigma(u^{N,n}_{s})\mathrm{d}W_s-\int_{0}^{t} S_{t-s}\sigma(u^{N}_{s})\mathrm{d}W_s \rightarrow 0\qquad ~{\rm as}~ n\rightarrow \infty,$$ in probability. Since $\|u^{N,n}-u^{N}\|_{\alpha,T}\rightarrow0, n \rightarrow \infty,$ a.s., we have the convergence of the integrals in probability on $\{\|u^{N}\|_{\alpha,T}\leq R\}$, where $R\geq 1$ is arbitrary, therefore the convergence holds on $\Omega$. This means that $u^N$ is a solution to $$\begin{aligned}
\label{map3}
u_t^{N}&=& S_t u_{0}+\int_{0}^{t} S_{t-s} f(u^{N}_s) d s+\int_{0}^{t} S_{t-s} \sigma(u^{N}_s) \mathrm{d}W_s\cr
&&+\int_{0}^{t} S_{t-s} g(u^{N}_s)\mathrm{d} B_s^{H,N}.\end{aligned}$$ It follows from Lemma \[lemboundNm\] that the processes $u^N$ and $u^M$ with $M\geq N$ coincide almost surely on the set $A_{N,T}=\{\Lambda^{0,T}_{\alpha,B^H} \leq N\}$. Hence, there exists 